<?xml version="1.0"?>
<div class="add">

<span class="kw">set </span><font color="Maroon" title="c3">W</font> =  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c1">C</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">E</font> =  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c1">C</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">S</font> =  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c1">C</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">N</font> =  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c1">C</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">G</font> =  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>( ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp;  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp;  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> ) ) &amp; ( $1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; $2 is   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> implies for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> holds <br/>( (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 implies ( ( <font color="Olive" title="b1">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies ( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) ) ) &amp; ( (  not <font color="Olive" title="b1">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> or  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> ) implies $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) ) &amp; (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> $1 implies $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) ) ) &amp; ( $1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; $2 is  not   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> implies $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) );<br/>
<a NAME="E1:35_1_1"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan8.html#D1" title="JORDAN8:def.1">JORDAN8:def 1</a></span>;<br/>
<a NAME="E2:35_1_1"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   ex <font color="Olive" title="b3">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>



<span class="kw">consider </span><font color="Maroon" title="c10">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E2:35_1_1_1"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font> &amp; <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp;  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c10">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan9.html#T30" target="_self" title="JORDAN9:th.30">Th30</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_1_1"/>
( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or ( <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c9">x</font> is   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) or ( <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c9">x</font> is  not   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_1_1_1"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">m</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">m</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]</span><br/><span class="kw">thus </span><a NAME="E2:35_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">m</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_1"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_1"><span class="lab"><font color="Green" title="E34">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_1_1_2"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span>
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">m</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">m</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>]</span><br/><span class="kw">thus </span><a NAME="E2:35_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">m</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_1"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_2"><span class="lab"><font color="Green" title="E34">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:35_1_1_1_1_3"/><span class="lab"><font color="Green" title="E34">A7</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_1_1_3"/><span class="lab"><font color="Green" title="E35">A8</font></span>: 
<font color="Maroon" title="c9">x</font> is   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c11">f</font> = <font color="Maroon" title="c9">x</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_1_1_3"><span class="lab"><font color="Green" title="E35">A8</font></span></a></span>;<br/><span class="kw">thus </span><a NAME="E4:35_1_1_1_1_3"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_1_1_3_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E36">A9</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:35_1_1_1_1_3_1_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1"/>
( ( <font color="Maroon" title="c11">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  not <font color="Maroon" title="c11">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> or  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E37">A10</font></span>: 
( <font color="Maroon" title="c11">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E38">A11</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A12</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E40">A13</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E5:35_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E41">A14</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">i1</font>, <font color="Maroon" title="c13">j1</font>, <font color="Maroon" title="c14">i2</font>, <font color="Maroon" title="c15">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:35_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E42">A15</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i1</font>,<font color="Maroon" title="c13">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:35_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E43">A16</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i1</font>,<font color="Maroon" title="c13">j1</font>
 <span class="kw">and </span><br/><a NAME="E9:35_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E44">A17</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i2</font>,<font color="Maroon" title="c15">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:35_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E45">A18</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i2</font>,<font color="Maroon" title="c15">j2</font>
 <span class="kw">and </span><br/><a NAME="E11:35_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E46">A19</font></span>: 
( ( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> ) or ( <font color="Maroon" title="c12">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> ) or ( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> ) or ( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>;<br/><a NAME="E12:35_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E47">A20</font></span>: 
<font color="Maroon" title="c12">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E13:35_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E48">A21</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E14:35_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E49">A22</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E15:35_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E50">A23</font></span>: 
<font color="Maroon" title="c13">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E16:35_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E51">A24</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E17:35_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E52">A25</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E18:35_1_1_1_1_3_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T35" title="NAT_D:th.35">NAT_D:35</a></span>;<br/><a NAME="E19:35_1_1_1_1_3_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A26</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E20:35_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E54">A27</font></span>: 
<font color="Maroon" title="c15">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E21:35_1_1_1_1_3_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A28</font></span>: 
<font color="Maroon" title="c15">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>;<br/><a NAME="E22:35_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E56">A29</font></span>: 
<font color="Maroon" title="c14">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E23:35_1_1_1_1_3_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A30</font></span>: 
<font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>;<br/><span class="kw">thus </span><a NAME="E24:35_1_1_1_1_3_1_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1"/>
( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> ) or (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E58">A31</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1"/>
( ( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> ) or ( <font color="Maroon" title="c12">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> ) or ( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> ) or ( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E59">A32</font></span>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c16">f1</font> = <font color="Maroon" title="c11">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">f1</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><span class="kw">take </span><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c15">j2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><span class="kw">thus </span><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">i19</font>, <font color="Maroon" title="c20">j19</font>, <font color="Maroon" title="c21">i29</font>, <font color="Maroon" title="c22">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or  not <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font> or  not <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font> or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E60">A33</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E61">A34</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E62">A35</font></span>: 
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E63">A36</font></span>: 
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) )</span><br/>

<a NAME="E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E64">A37</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E62">A35</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<a NAME="E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A38</font></span>: 
<font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E60">A33</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E7:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E66">A39</font></span>: 
<font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E60">A33</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E64">A37</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E67">A40</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E63">A36</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<a NAME="E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A41</font></span>: 
<font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E61">A34</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E69">A42</font></span>: 
<font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E61">A34</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E67">A40</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E11:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E60">A33</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E64">A37</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E68">A41</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A42</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A43</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E70">A43</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c13">j1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E59">A32</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E15:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="ref" href="jordan8.html#T19" title="JORDAN8:th.19">JORDAN8:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="txt" href="#E20:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="txt" href="#E16:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E68">A41</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A42</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><br/><span class="kw">thus </span><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E68">A41</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A42</font></span></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_2"/>
( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E59">A32</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E65">A38</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E68">A41</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_3"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_3"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E59">A32</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E65">A38</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E68">A41</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_4"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_4"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E59">A32</font></span></a>, <a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E66">A39</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A42</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">f1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E58">A31</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E59">A44</font></span>: 
( <font color="Maroon" title="c12">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c16">f1</font> = <font color="Maroon" title="c11">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">f1</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c14">i2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><span class="kw">take </span><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c15">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><span class="kw">thus </span><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">i19</font>, <font color="Maroon" title="c20">j19</font>, <font color="Maroon" title="c21">i29</font>, <font color="Maroon" title="c22">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or  not <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font> or  not <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font> or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E60">A45</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E61">A46</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E62">A47</font></span>: 
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E63">A48</font></span>: 
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) )</span><br/>

<a NAME="E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E64">A49</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E63">A48</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<a NAME="E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A50</font></span>: 
<font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E7:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E66">A51</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<a NAME="E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A52</font></span>: 
<font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E68">A53</font></span>: 
<font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E64">A49</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"/>
( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E11:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E66">A51</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E65">A50</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E68">A53</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_1"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E59">A44</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E67">A52</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E65">A50</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2"/>
( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2_1"/>
<font color="Maroon" title="c15">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="nat_d.html#T36" title="NAT_D:th.36">NAT_D:36</a></span>;<br/><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">i1</font>,<span class="p1">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E59">A44</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>;<br/><a NAME="E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">i1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T234" title="XREAL_1:th.234">XREAL_1:234</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E47">A20</font></span></a>, <a class="ref" href="jordan8.html#T20" title="JORDAN8:th.20">JORDAN8:20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="txt" href="#E22:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="txt" href="#E21:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E55">A28</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E65">A50</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E68">A53</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><br/><span class="kw">thus </span><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E65">A50</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E68">A53</font></span></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_3"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_3"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E59">A44</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E67">A52</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E65">A50</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_4"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_4"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E59">A44</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E67">A52</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E65">A50</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">f1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E58">A31</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E59">A54</font></span>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c16">f1</font> = <font color="Maroon" title="c11">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">f1</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c14">i2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><span class="kw">take </span><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><span class="kw">thus </span><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1"/>
<font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">i19</font>, <font color="Maroon" title="c20">j19</font>, <font color="Maroon" title="c21">i29</font>, <font color="Maroon" title="c22">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or  not <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font> or  not <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font> or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><span class="lab"><font color="Green" title="E60">A55</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><span class="lab"><font color="Green" title="E61">A56</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><span class="lab"><font color="Green" title="E62">A57</font></span>: 
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><span class="lab"><font color="Green" title="E63">A58</font></span>: 
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) )</span><br/>

<a NAME="E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><span class="lab"><font color="Green" title="E64">A59</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E63">A58</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<a NAME="E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A60</font></span>: 
<font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E61">A56</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E7:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><span class="lab"><font color="Green" title="E66">A61</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E62">A57</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<a NAME="E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A62</font></span>: 
<font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E60">A55</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><span class="lab"><font color="Green" title="E68">A63</font></span>: 
<font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E61">A56</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E64">A59</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/>
( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E11:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E60">A55</font></span></a>, <a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E66">A61</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E65">A60</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E68">A63</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_1"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E59">A54</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E67">A62</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E65">A60</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_2"/>
( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E59">A54</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E67">A62</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E65">A60</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3_1"/>
<font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A64</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3_1"/>
<font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E20:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3_1"><span class="lab"><font color="Green" title="E69">A64</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c14">i2</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E59">A54</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="ref" href="jordan8.html#T18" title="JORDAN8:th.18">JORDAN8:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E14:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="txt" href="#E22:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="txt" href="#E13:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E48">A21</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E65">A60</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E68">A63</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><span class="kw">thus </span><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3"/>
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E65">A60</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E68">A63</font></span></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_4"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_4"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E59">A54</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E67">A62</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E65">A60</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">f1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E58">A31</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E59">A65</font></span>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c16">f1</font> = <font color="Maroon" title="c11">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">f1</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><span class="kw">take </span><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c15">j2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><span class="kw">thus </span><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1"/>
<font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">i19</font>, <font color="Maroon" title="c20">j19</font>, <font color="Maroon" title="c21">i29</font>, <font color="Maroon" title="c22">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or  not <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font> or  not <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font> or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><span class="lab"><font color="Green" title="E60">A66</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><span class="lab"><font color="Green" title="E61">A67</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><span class="lab"><font color="Green" title="E62">A68</font></span>: 
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><span class="lab"><font color="Green" title="E63">A69</font></span>: 
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) )</span><br/>

<a NAME="E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><span class="lab"><font color="Green" title="E64">A70</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E62">A68</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<a NAME="E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A71</font></span>: 
<font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E60">A66</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E7:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><span class="lab"><font color="Green" title="E66">A72</font></span>: 
<font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E60">A66</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E64">A70</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><span class="lab"><font color="Green" title="E67">A73</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E63">A69</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<a NAME="E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A74</font></span>: 
<font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E61">A67</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><span class="lab"><font color="Green" title="E69">A75</font></span>: 
<font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E61">A67</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E67">A73</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1"/>
( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E11:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E60">A66</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E64">A70</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E68">A74</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E69">A75</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_1"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E59">A65</font></span></a>, <a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E66">A72</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E69">A75</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_2"/>
( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E59">A65</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E65">A71</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E68">A74</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_3"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_3"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E59">A65</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E65">A71</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E68">A74</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4_1"/>
<font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="nat_d.html#T36" title="NAT_D:th.36">NAT_D:36</a></span>;<br/><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E59">A65</font></span></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></span>;<br/><a NAME="E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c15">j2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T234" title="XREAL_1:th.234">XREAL_1:234</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E20:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="ref" href="jordan8.html#T21" title="JORDAN8:th.21">JORDAN8:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="txt" href="#E20:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="txt" href="#E23:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E57">A30</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E68">A74</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E69">A75</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><br/><span class="kw">thus </span><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4"/>
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E68">A74</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E69">A75</font></span></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">f1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E58">A31</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E58">A76</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1"/>
( ( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> ) or ( <font color="Maroon" title="c12">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> ) or ( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> ) or ( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E59">A77</font></span>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c16">f1</font> = <font color="Maroon" title="c11">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">f1</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c14">i2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><span class="kw">take </span><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><span class="kw">thus </span><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">i19</font>, <font color="Maroon" title="c20">j19</font>, <font color="Maroon" title="c21">i29</font>, <font color="Maroon" title="c22">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or  not <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font> or  not <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font> or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E60">A78</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E61">A79</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E62">A80</font></span>: 
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E63">A81</font></span>: 
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) )</span><br/>

<a NAME="E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E64">A82</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E62">A80</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<a NAME="E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A83</font></span>: 
<font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E60">A78</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E7:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E66">A84</font></span>: 
<font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E60">A78</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E64">A82</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E67">A85</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E63">A81</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<a NAME="E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A86</font></span>: 
<font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E61">A79</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E69">A87</font></span>: 
<font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E61">A79</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E67">A85</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1"/>
( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E11:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E60">A78</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E64">A82</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E68">A86</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A87</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A88</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E20:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E70">A88</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">i1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A76</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E59">A77</font></span></a>, <a class="ref" href="gobrd13.html#T36" title="GOBRD13:th.36">GOBRD13:36</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E47">A20</font></span></a>, <a class="ref" href="jordan8.html#T18" title="JORDAN8:th.18">JORDAN8:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E14:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="txt" href="#E22:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="txt" href="#E13:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E48">A21</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E68">A86</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A87</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><span class="kw">thus </span><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E68">A86</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A87</font></span></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_2"/>
( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E59">A77</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E65">A83</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E68">A86</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_3"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_3"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E59">A77</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E65">A83</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E68">A86</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_4"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_4"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E59">A77</font></span></a>, <a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E66">A84</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A87</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">f1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A76</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E59">A89</font></span>: 
( <font color="Maroon" title="c12">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c16">f1</font> = <font color="Maroon" title="c11">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">f1</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><span class="kw">take </span><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c15">j2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><span class="kw">thus </span><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">i19</font>, <font color="Maroon" title="c20">j19</font>, <font color="Maroon" title="c21">i29</font>, <font color="Maroon" title="c22">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or  not <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font> or  not <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font> or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E60">A90</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E61">A91</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E62">A92</font></span>: 
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E63">A93</font></span>: 
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) )</span><br/>

<a NAME="E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E64">A94</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E63">A93</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<a NAME="E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A95</font></span>: 
<font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E61">A91</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E7:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E66">A96</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E62">A92</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<a NAME="E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A97</font></span>: 
<font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E60">A90</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E68">A98</font></span>: 
<font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E61">A91</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E64">A94</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/>
( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E11:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E60">A90</font></span></a>, <a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E66">A96</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E65">A95</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E68">A98</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_1"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E59">A89</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E67">A97</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E65">A95</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2"/>
( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2_1"/>
<font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A99</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2_1"/>
<font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E69">A99</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A76</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E59">A89</font></span></a>, <a class="ref" href="gobrd13.html#T38" title="GOBRD13:th.38">GOBRD13:38</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E15:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="ref" href="jordan8.html#T19" title="JORDAN8:th.19">JORDAN8:19</a>, <a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="txt" href="#E20:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="txt" href="#E16:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E65">A95</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E68">A98</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><br/><span class="kw">thus </span><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E65">A95</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E68">A98</font></span></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_3"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_3"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E59">A89</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E67">A97</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E65">A95</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_4"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_4"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E59">A89</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E67">A97</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E65">A95</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">f1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A76</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E59">A100</font></span>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c16">f1</font> = <font color="Maroon" title="c11">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">f1</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><span class="kw">take </span><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c15">j2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><span class="kw">thus </span><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1"/>
<font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">i19</font>, <font color="Maroon" title="c20">j19</font>, <font color="Maroon" title="c21">i29</font>, <font color="Maroon" title="c22">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or  not <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font> or  not <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font> or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><span class="lab"><font color="Green" title="E60">A101</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><span class="lab"><font color="Green" title="E61">A102</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><span class="lab"><font color="Green" title="E62">A103</font></span>: 
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><span class="lab"><font color="Green" title="E63">A104</font></span>: 
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) )</span><br/>

<a NAME="E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><span class="lab"><font color="Green" title="E64">A105</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E63">A104</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<a NAME="E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A106</font></span>: 
<font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E61">A102</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E7:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><span class="lab"><font color="Green" title="E66">A107</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E62">A103</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<a NAME="E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A108</font></span>: 
<font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E60">A101</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><span class="lab"><font color="Green" title="E68">A109</font></span>: 
<font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E61">A102</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E64">A105</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"/>
( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E11:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E60">A101</font></span></a>, <a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E66">A107</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E65">A106</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E68">A109</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_1"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E59">A100</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E67">A108</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E65">A106</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_2"/>
( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E59">A100</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E67">A108</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E65">A106</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3_1"/>
<font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="nat_d.html#T36" title="NAT_D:th.36">NAT_D:36</a></span>;<br/><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c13">j1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A76</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E59">A100</font></span></a>, <a class="ref" href="gobrd13.html#T40" title="GOBRD13:th.40">GOBRD13:40</a></span>;<br/><a NAME="E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c13">j1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T234" title="XREAL_1:th.234">XREAL_1:234</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E15:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="ref" href="jordan8.html#T21" title="JORDAN8:th.21">JORDAN8:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="txt" href="#E20:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="txt" href="#E23:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E57">A30</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E65">A106</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E68">A109</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><br/><span class="kw">thus </span><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3"/>
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E65">A106</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E68">A109</font></span></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_4"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_4"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E59">A100</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E67">A108</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E65">A106</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">f1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A76</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"/><span class="lab"><font color="Green" title="E59">A110</font></span>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c16">f1</font> = <font color="Maroon" title="c11">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">f1</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c14">i2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><span class="kw">take </span><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c15">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><span class="kw">thus </span><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1"/>
<font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">i19</font>, <font color="Maroon" title="c20">j19</font>, <font color="Maroon" title="c21">i29</font>, <font color="Maroon" title="c22">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or  not <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font> or  not <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font> or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><span class="lab"><font color="Green" title="E60">A111</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><span class="lab"><font color="Green" title="E61">A112</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><span class="lab"><font color="Green" title="E62">A113</font></span>: 
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><span class="lab"><font color="Green" title="E63">A114</font></span>: 
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) )</span><br/>

<a NAME="E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><span class="lab"><font color="Green" title="E64">A115</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E62">A113</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<a NAME="E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A116</font></span>: 
<font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E60">A111</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E7:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><span class="lab"><font color="Green" title="E66">A117</font></span>: 
<font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E60">A111</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E64">A115</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><span class="lab"><font color="Green" title="E67">A118</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E63">A114</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<a NAME="E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A119</font></span>: 
<font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E61">A112</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><span class="lab"><font color="Green" title="E69">A120</font></span>: 
<font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E61">A112</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E67">A118</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"/>
( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E11:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E60">A111</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E64">A115</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E68">A119</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E69">A120</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_1"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E59">A110</font></span></a>, <a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E66">A117</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E69">A120</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_2"/>
( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E59">A110</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E65">A116</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E68">A119</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_3"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_3"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E59">A110</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E65">A116</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E68">A119</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4_1"/>
<font color="Maroon" title="c15">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="nat_d.html#T36" title="NAT_D:th.36">NAT_D:36</a></span>;<br/><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A76</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E59">A110</font></span></a>, <a class="ref" href="gobrd13.html#T42" title="GOBRD13:th.42">GOBRD13:42</a></span>;<br/><a NAME="E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T234" title="XREAL_1:th.234">XREAL_1:234</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E47">A20</font></span></a>, <a class="ref" href="jordan8.html#T20" title="JORDAN8:th.20">JORDAN8:20</a>, <a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="txt" href="#E22:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="txt" href="#E21:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E55">A28</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E68">A119</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E69">A120</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><br/><span class="kw">thus </span><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4"/>
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E68">A119</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E69">A120</font></span></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">f1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A76</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E58">A121</font></span>: 
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1"/>
( ( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> ) or ( <font color="Maroon" title="c12">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> ) or ( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> ) or ( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E59">A122</font></span>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c16">f1</font> = <font color="Maroon" title="c11">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">f1</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><span class="kw">take </span><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c15">j2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><span class="kw">thus </span><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1"/>
<font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">i19</font>, <font color="Maroon" title="c20">j19</font>, <font color="Maroon" title="c21">i29</font>, <font color="Maroon" title="c22">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or  not <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font> or  not <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font> or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E60">A123</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E61">A124</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E62">A125</font></span>: 
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E63">A126</font></span>: 
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) )</span><br/>

<a NAME="E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E64">A127</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E62">A125</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<a NAME="E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A128</font></span>: 
<font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E60">A123</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E7:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E66">A129</font></span>: 
<font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E60">A123</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E64">A127</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E67">A130</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E63">A126</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<a NAME="E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A131</font></span>: 
<font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E61">A124</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E69">A132</font></span>: 
<font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E61">A124</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E67">A130</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1"/>
( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E11:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E60">A123</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E64">A127</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E68">A131</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A132</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="nat_d.html#T36" title="NAT_D:th.36">NAT_D:36</a></span>;<br/><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E58">A121</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E59">A122</font></span></a>, <a class="ref" href="gobrd13.html#T35" title="GOBRD13:th.35">GOBRD13:35</a></span>;<br/><a NAME="E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c15">j2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T234" title="XREAL_1:th.234">XREAL_1:234</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E20:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="ref" href="jordan8.html#T21" title="JORDAN8:th.21">JORDAN8:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="txt" href="#E20:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="txt" href="#E23:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E57">A30</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E68">A131</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A132</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><br/><span class="kw">thus </span><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E68">A131</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A132</font></span></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_2"/>
( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E59">A122</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E65">A128</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E68">A131</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_3"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_3"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E59">A122</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E65">A128</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E68">A131</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_4"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_4"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E59">A122</font></span></a>, <a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E66">A129</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A132</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">f1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E58">A121</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E59">A133</font></span>: 
( <font color="Maroon" title="c12">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c16">f1</font> = <font color="Maroon" title="c11">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">f1</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c14">i2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><span class="kw">take </span><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><span class="kw">thus </span><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1"/>
<font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">i19</font>, <font color="Maroon" title="c20">j19</font>, <font color="Maroon" title="c21">i29</font>, <font color="Maroon" title="c22">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or  not <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font> or  not <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font> or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><span class="lab"><font color="Green" title="E60">A134</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><span class="lab"><font color="Green" title="E61">A135</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><span class="lab"><font color="Green" title="E62">A136</font></span>: 
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><span class="lab"><font color="Green" title="E63">A137</font></span>: 
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) )</span><br/>

<a NAME="E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><span class="lab"><font color="Green" title="E64">A138</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E63">A137</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<a NAME="E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A139</font></span>: 
<font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E61">A135</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E7:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><span class="lab"><font color="Green" title="E66">A140</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E62">A136</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<a NAME="E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A141</font></span>: 
<font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E60">A134</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><span class="lab"><font color="Green" title="E68">A142</font></span>: 
<font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E61">A135</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E64">A138</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"/>
( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E11:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E60">A134</font></span></a>, <a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E66">A140</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E65">A139</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E68">A142</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_1"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E59">A133</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E67">A141</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E65">A139</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2"/>
( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2_1"/>
<font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A143</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2_1"/>
<font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E20:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E69">A143</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c14">i2</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E58">A121</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E59">A133</font></span></a>, <a class="ref" href="gobrd13.html#T37" title="GOBRD13:th.37">GOBRD13:37</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="ref" href="jordan8.html#T18" title="JORDAN8:th.18">JORDAN8:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E14:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="txt" href="#E22:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="txt" href="#E13:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E48">A21</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E65">A139</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E68">A142</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><span class="kw">thus </span><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E65">A139</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E68">A142</font></span></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_3"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_3"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E59">A133</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E67">A141</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E65">A139</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_4"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_4"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E59">A133</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E67">A141</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E65">A139</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">f1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E58">A121</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E59">A144</font></span>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c16">f1</font> = <font color="Maroon" title="c11">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">f1</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c14">i2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><span class="kw">take </span><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c15">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><span class="kw">thus </span><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1"/>
<font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">i19</font>, <font color="Maroon" title="c20">j19</font>, <font color="Maroon" title="c21">i29</font>, <font color="Maroon" title="c22">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or  not <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font> or  not <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font> or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><span class="lab"><font color="Green" title="E60">A145</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><span class="lab"><font color="Green" title="E61">A146</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><span class="lab"><font color="Green" title="E62">A147</font></span>: 
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><span class="lab"><font color="Green" title="E63">A148</font></span>: 
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) )</span><br/>

<a NAME="E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><span class="lab"><font color="Green" title="E64">A149</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E63">A148</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<a NAME="E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A150</font></span>: 
<font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E61">A146</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E7:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><span class="lab"><font color="Green" title="E66">A151</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E62">A147</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<a NAME="E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A152</font></span>: 
<font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E60">A145</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><span class="lab"><font color="Green" title="E68">A153</font></span>: 
<font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E61">A146</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E64">A149</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1"/>
( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E11:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E60">A145</font></span></a>, <a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E66">A151</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E65">A150</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E68">A153</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_1"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E59">A144</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E67">A152</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E65">A150</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_2"/>
( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E59">A144</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E67">A152</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E65">A150</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3_1"/>
<font color="Maroon" title="c15">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="nat_d.html#T36" title="NAT_D:th.36">NAT_D:36</a></span>;<br/><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E58">A121</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E59">A144</font></span></a>, <a class="ref" href="gobrd13.html#T39" title="GOBRD13:th.39">GOBRD13:39</a></span>;<br/><a NAME="E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T234" title="XREAL_1:th.234">XREAL_1:234</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="ref" href="jordan8.html#T20" title="JORDAN8:th.20">JORDAN8:20</a>, <a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="txt" href="#E22:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="txt" href="#E21:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E55">A28</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E65">A150</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E68">A153</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><br/><span class="kw">thus </span><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3"/>
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E65">A150</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E68">A153</font></span></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_4"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_4"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E59">A144</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E67">A152</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E65">A150</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">f1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E58">A121</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"/><span class="lab"><font color="Green" title="E59">A154</font></span>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c16">f1</font> = <font color="Maroon" title="c11">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">f1</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><span class="kw">take </span><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c15">j2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><span class="kw">thus </span><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1"/>
<font color="Maroon" title="c16">f1</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">i19</font>, <font color="Maroon" title="c20">j19</font>, <font color="Maroon" title="c21">i29</font>, <font color="Maroon" title="c22">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or  not <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font> or  not <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font> or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><span class="lab"><font color="Green" title="E60">A155</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><span class="lab"><font color="Green" title="E61">A156</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><span class="lab"><font color="Green" title="E62">A157</font></span>: 
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><span class="lab"><font color="Green" title="E63">A158</font></span>: 
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> ) )</span><br/>

<a NAME="E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><span class="lab"><font color="Green" title="E64">A159</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E62">A157</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<a NAME="E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A160</font></span>: 
<font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E60">A155</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E7:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><span class="lab"><font color="Green" title="E66">A161</font></span>: 
<font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E60">A155</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E64">A159</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><span class="lab"><font color="Green" title="E67">A162</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E63">A158</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<a NAME="E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A163</font></span>: 
<font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E61">A156</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><span class="lab"><font color="Green" title="E69">A164</font></span>: 
<font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E61">A156</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E67">A162</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"/>
( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E11:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E60">A155</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E64">A159</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E68">A163</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E69">A164</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_1"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E59">A154</font></span></a>, <a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E66">A161</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E69">A164</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_2"/>
( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E59">A154</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E65">A160</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E68">A163</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_3"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_3"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E59">A154</font></span></a>, <a class="txt" href="#E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E65">A160</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E68">A163</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4"><span class="kw">case </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4"/>
( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4_1"/>
<font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A165</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4_1"/>
<font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4_1"><span class="lab"><font color="Green" title="E70">A165</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E5:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E2:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E7:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E58">A121</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E59">A154</font></span></a>, <a class="ref" href="gobrd13.html#T41" title="GOBRD13:th.41">GOBRD13:41</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E20:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="ref" href="jordan8.html#T19" title="JORDAN8:th.19">JORDAN8:19</a>, <a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="txt" href="#E20:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="txt" href="#E16:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E68">A163</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E69">A164</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font></span><br/><span class="kw">thus </span><a NAME="E4:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4"/>
<font color="Maroon" title="c16">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="txt" href="#E9:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E68">A163</font></span></a>, <a class="txt" href="#E10:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E69">A164</font></span></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">f1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E58">A121</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_1_1_3_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E37">A166</font></span>: 
(  not <font color="Maroon" title="c11">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> or  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span>
<font color="Maroon" title="c11">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]</span><br/><span class="kw">thus </span><a NAME="E2:35_1_1_1_1_3_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E37">A166</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E36">A167</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]</span><br/><span class="kw">thus </span><a NAME="E2:35_1_1_1_1_3_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_3"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E36">A167</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_1_1_4"/><span class="lab"><font color="Green" title="E34">A168</font></span>: 
( <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c9">x</font> is  not   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]</span><br/><span class="kw">thus </span><a NAME="E2:35_1_1_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">x</font>, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1_1_4"><span class="lab"><font color="Green" title="E34">A168</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c8">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E4:35_1_1"/><span class="lab"><font color="Green" title="E33">A169</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">and </span><br/><a NAME="E5:35_1_1"/><span class="lab"><font color="Green" title="E34">A170</font></span>: 
<font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E6:35_1_1"/><span class="lab"><font color="Green" title="E35">A171</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>,<font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="recdef_1.html#S1" title="RECDEF_1:sch.1">RECDEF_1:sch 1</a>(<a class="txt" href="#E2:35_1_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a>);<br/></span>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span><font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 is   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>
<a NAME="E7:35_1_1"/><span class="lab"><font color="Green" title="E36">A172</font></span>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
;<br/>
<a NAME="E8:35_1_1"/><span class="lab"><font color="Green" title="E37">A173</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1_1_2"/><span class="lab"><font color="Green" title="E38">A174</font></span>: 
<font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> is   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E2:35_1_1_2"/><span class="lab"><font color="Green" title="E39">A175</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_2_1"/>
( <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_2_1_1"/>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_2_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_2"><span class="lab"><font color="Green" title="E39">A175</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_2_1_2"/>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_2_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_2"><span class="lab"><font color="Green" title="E39">A175</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_2_1_3"/><span class="lab"><font color="Green" title="E40">A176</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:35_1_1_2_1_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_2"><span class="lab"><font color="Green" title="E38">A174</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_2_1_3_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_2_1_3_1_1_1"/><span class="lab"><font color="Green" title="E41">A177</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:35_1_1_2_1_3_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2_1_3_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_1_3_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_2_1_3_1_1_1_1_1"/>
( ( <font color="Maroon" title="c10">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c10">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  not <font color="Maroon" title="c10">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> or  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c10">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_1_3_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_2_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E42">A178</font></span>: 
( <font color="Maroon" title="c10">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c10">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:35_1_1_2_1_3_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A179</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c10">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c10">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a>, <a class="txt" href="#E1:35_1_1_2_1_3"><span class="lab"><font color="Green" title="E40">A176</font></span></a>, <a class="txt" href="#E1:35_1_1_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E41">A177</font></span></a></span>;<br/><a NAME="E3:35_1_1_2_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E44">A180</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c10">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c10">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c10">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a>, <a class="txt" href="#E1:35_1_1_2_1_3"><span class="lab"><font color="Green" title="E40">A176</font></span></a>, <a class="txt" href="#E1:35_1_1_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E41">A177</font></span></a>, <a class="txt" href="#E1:35_1_1_2_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A178</font></span></a></span>;<br/><a NAME="E4:35_1_1_2_1_3_1_1_1_1_1_1"/>
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c10">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c10">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c10">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a>, <a class="txt" href="#E1:35_1_1_2_1_3"><span class="lab"><font color="Green" title="E40">A176</font></span></a>, <a class="txt" href="#E1:35_1_1_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E41">A177</font></span></a>, <a class="txt" href="#E1:35_1_1_2_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A178</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:35_1_1_2_1_3_1_1_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_2_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A180</font></span></a>, <a class="txt" href="#E2:35_1_1_2_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A179</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_1_3_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_2_1_3_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E42">A181</font></span>: 
(  not <font color="Maroon" title="c10">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> or  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c10">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:35_1_1_2_1_3_1_1_1_1_1_2"/>
<font color="Maroon" title="c10">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> is   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 ;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_2_1_3_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a>, <a class="txt" href="#E1:35_1_1_2_1_3"><span class="lab"><font color="Green" title="E40">A176</font></span></a>, <a class="txt" href="#E1:35_1_1_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E41">A177</font></span></a>, <a class="txt" href="#E1:35_1_1_2_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E42">A181</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_2_1_3_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_2_1_3_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a>, <a class="txt" href="#E7:35_1_1"><span class="lab"><font color="Green" title="E36">A172</font></span></a>, <a class="txt" href="#E1:35_1_1_2_1_3"><span class="lab"><font color="Green" title="E40">A176</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E9:35_1_1"/><span class="lab"><font color="Green" title="E38">A182</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1"><span class="lab"><font color="Green" title="E34">A170</font></span></a>, <a class="txt" href="#E7:35_1_1"><span class="lab"><font color="Green" title="E36">A172</font></span></a></span>;<br/>
<a NAME="E10:35_1_1"/><span class="lab"><font color="Green" title="E39">A183</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E9:35_1_1"><span class="lab"><font color="Green" title="E38">A182</font></span></a>, <a class="txt" href="#E8:35_1_1"><span class="lab"><font color="Green" title="E37">A173</font></span></a>);<br/></span>
<a NAME="E11:35_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">F</font> or <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1_1_3"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span><br/>

<a NAME="E2:35_1_1_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">F</font> &amp; <font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E3:35_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">y</font> is   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1"><span class="lab"><font color="Green" title="E33">A169</font></span></a>, <a class="txt" href="#E10:35_1_1"><span class="lab"><font color="Green" title="E39">A183</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:35_1_1_3"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">F</font> = <font color="Maroon" title="c8">F</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1"><span class="lab"><font color="Green" title="E33">A169</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1;<br/>
<a NAME="E13:35_1_1"/><span class="lab"><font color="Green" title="E40">A184</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">k</font>] implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1_1_4"/><span class="lab"><font color="Green" title="E41">A185</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E2:35_1_1_4"/><span class="lab"><font color="Green" title="E42">A186</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_4_1"/>
( <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_4_1_1"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_4_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_4"><span class="lab"><font color="Green" title="E42">A186</font></span></a>, <a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_4_1_2"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_4_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_4"><span class="lab"><font color="Green" title="E42">A186</font></span></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_4_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_4_1_3"/><span class="lab"><font color="Green" title="E43">A187</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:35_1_1_4_1_3"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_4_1_3_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_4_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_4_1_3_1_1"/>
( ( <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  not <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> or  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_4_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_4_1_3_1_1_1"/><span class="lab"><font color="Green" title="E44">A188</font></span>: 
( <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:35_1_1_4_1_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A189</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a>, <a class="txt" href="#E1:35_1_1_4"><span class="lab"><font color="Green" title="E41">A185</font></span></a>, <a class="txt" href="#E1:35_1_1_4_1_3"><span class="lab"><font color="Green" title="E43">A187</font></span></a></span>;<br/><a NAME="E3:35_1_1_4_1_3_1_1_1"/><span class="lab"><font color="Green" title="E46">A190</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a>, <a class="txt" href="#E1:35_1_1_4"><span class="lab"><font color="Green" title="E41">A185</font></span></a>, <a class="txt" href="#E1:35_1_1_4_1_3"><span class="lab"><font color="Green" title="E43">A187</font></span></a>, <a class="txt" href="#E1:35_1_1_4_1_3_1_1_1"><span class="lab"><font color="Green" title="E44">A188</font></span></a></span>;<br/><a NAME="E4:35_1_1_4_1_3_1_1_1"/>
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a>, <a class="txt" href="#E1:35_1_1_4"><span class="lab"><font color="Green" title="E41">A185</font></span></a>, <a class="txt" href="#E1:35_1_1_4_1_3"><span class="lab"><font color="Green" title="E43">A187</font></span></a>, <a class="txt" href="#E1:35_1_1_4_1_3_1_1_1"><span class="lab"><font color="Green" title="E44">A188</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:35_1_1_4_1_3_1_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_4"><span class="lab"><font color="Green" title="E41">A185</font></span></a>, <a class="txt" href="#E3:35_1_1_4_1_3_1_1_1"><span class="lab"><font color="Green" title="E46">A190</font></span></a>, <a class="txt" href="#E2:35_1_1_4_1_3_1_1_1"><span class="lab"><font color="Green" title="E45">A189</font></span></a>, <a class="ref" href="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_4_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_4_1_3_1_1_2"/>
(  not <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> or  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:35_1_1_4_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a>, <a class="txt" href="#E1:35_1_1_4"><span class="lab"><font color="Green" title="E41">A185</font></span></a>, <a class="txt" href="#E1:35_1_1_4_1_3"><span class="lab"><font color="Green" title="E43">A187</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_4_1_3_1_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_4"><span class="lab"><font color="Green" title="E41">A185</font></span></a>, <a class="ref" href="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E14:35_1_1"/><span class="lab"><font color="Green" title="E41">A191</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1"><span class="lab"><font color="Green" title="E34">A170</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></span>;<br/>
<a NAME="E15:35_1_1"/><span class="lab"><font color="Green" title="E42">A192</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E14:35_1_1"><span class="lab"><font color="Green" title="E41">A191</font></span></a>, <a class="txt" href="#E13:35_1_1"><span class="lab"><font color="Green" title="E40">A184</font></span></a>);<br/></span>
<div><span class="lab"><font color="Green" title="E43">A193</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; ( for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> holds <br/>(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) ) &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 implies for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">j1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font> holds <br/>( ( <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; ( <font color="Olive" title="b1">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; ( <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b4">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; ( <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b4">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Olive" title="b1">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Olive" title="b1">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b4">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_1_5"/><span class="lab"><font color="Green" title="E44">A194</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_5"/><span class="lab"><font color="Green" title="E45">A195</font></span>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> holds <br/>(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <span class="kw">and </span><br/><a NAME="E3:35_1_1_5"/><span class="lab"><font color="Green" title="E46">A196</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">j1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font> holds <br/>( ( <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; ( <font color="Olive" title="b1">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; ( <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b4">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; ( <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b4">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Olive" title="b1">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Olive" title="b1">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b4">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) )</span><br/><a NAME="E4:35_1_1_5"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a></span>;<br/><a NAME="E5:35_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A197</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_5"><span class="lab"><font color="Green" title="E46">A196</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/><a NAME="E6:35_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A198</font></span>: 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_5"><span class="lab"><font color="Green" title="E45">A195</font></span></a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c11">i1</font>, <font color="Maroon" title="c12">j1</font>, <font color="Maroon" title="c13">i2</font>, <font color="Maroon" title="c14">j2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i1</font>,<font color="Maroon" title="c12">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i1</font>,<font color="Maroon" title="c12">j1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<font color="Maroon" title="c14">j2</font> implies ( ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E7:35_1_1_5"/><span class="lab"><font color="Green" title="E49">A199</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i1</font>,<font color="Maroon" title="c12">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:35_1_1_5"/><span class="lab"><font color="Green" title="E50">A200</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i1</font>,<font color="Maroon" title="c12">j1</font>
 <span class="kw">and </span><br/><a NAME="E9:35_1_1_5"/><span class="lab"><font color="Green" title="E51">A201</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:35_1_1_5"/><span class="lab"><font color="Green" title="E52">A202</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<font color="Maroon" title="c14">j2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) )</span><br/><a NAME="E11:35_1_1_5"/><span class="lab"><font color="Green" title="E53">A203</font></span>: 
<font color="Maroon" title="c13">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_5"><span class="lab"><font color="Green" title="E51">A201</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E12:35_1_1_5"/><span class="lab"><font color="Green" title="E54">A204</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E13:35_1_1_5"/><span class="lab"><font color="Green" title="E55">A205</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_5"><span class="lab"><font color="Green" title="E51">A201</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E14:35_1_1_5"/><span class="lab"><font color="Green" title="E56">A206</font></span>: 
<font color="Maroon" title="c14">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_5"><span class="lab"><font color="Green" title="E51">A201</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E15:35_1_1_5"/><span class="lab"><font color="Green" title="E57">A207</font></span>: 
<font color="Maroon" title="c12">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_5"><span class="lab"><font color="Green" title="E49">A199</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_5_1"/><span class="lab"><font color="Green" title="E58">A208</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_5_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_5_1_1"/>
<font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A209</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:35_1_1_5_1_1"/>
<font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:35_1_1_5"><span class="lab"><font color="Green" title="E53">A203</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E4:35_1_1_5_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_5_1_1"><span class="lab"><font color="Green" title="E59">A209</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:35_1_1_5_1_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c12">j1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_5"><span class="lab"><font color="Green" title="E44">A194</font></span></a>, <a class="txt" href="#E7:35_1_1_5"><span class="lab"><font color="Green" title="E49">A199</font></span></a>, <a class="txt" href="#E8:35_1_1_5"><span class="lab"><font color="Green" title="E50">A200</font></span></a>, <a class="txt" href="#E9:35_1_1_5"><span class="lab"><font color="Green" title="E51">A201</font></span></a>, <a class="txt" href="#E10:35_1_1_5"><span class="lab"><font color="Green" title="E52">A202</font></span></a>, <a class="txt" href="#E5:35_1_1_5"><span class="lab"><font color="Green" title="E47">A197</font></span></a>, <a class="txt" href="#E6:35_1_1_5"><span class="lab"><font color="Green" title="E48">A198</font></span></a>, <a class="txt" href="#E1:35_1_1_5_1"><span class="lab"><font color="Green" title="E58">A208</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_5_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E15:35_1_1_5"><span class="lab"><font color="Green" title="E57">A207</font></span></a>, <a class="ref" href="jordan8.html#T19" title="JORDAN8:th.19">JORDAN8:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_5_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:35_1_1_5"><span class="lab"><font color="Green" title="E55">A205</font></span></a>, <a class="txt" href="#E14:35_1_1_5"><span class="lab"><font color="Green" title="E56">A206</font></span></a>, <a class="txt" href="#E12:35_1_1_5"><span class="lab"><font color="Green" title="E54">A204</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><a NAME="E17:35_1_1_5"/><span class="lab"><font color="Green" title="E58">A210</font></span>: 
<font color="Maroon" title="c11">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_5"><span class="lab"><font color="Green" title="E49">A199</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E18:35_1_1_5"/><span class="lab"><font color="Green" title="E59">A211</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_5"><span class="lab"><font color="Green" title="E51">A201</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E19:35_1_1_5"/><span class="lab"><font color="Green" title="E60">A212</font></span>: 
<font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:35_1_1_5"><span class="lab"><font color="Green" title="E56">A206</font></span></a>, <a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>;<br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_5_2"/><span class="lab"><font color="Green" title="E61">A213</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_5_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_5_2_1"/>
<font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_5_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="nat_d.html#T36" title="NAT_D:th.36">NAT_D:36</a></span>;<br/><a NAME="E3:35_1_1_5_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:35_1_1_5"><span class="lab"><font color="Green" title="E55">A205</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:35_1_1_5_2_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i1</font>,<span class="p1">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_5"><span class="lab"><font color="Green" title="E44">A194</font></span></a>, <a class="txt" href="#E7:35_1_1_5"><span class="lab"><font color="Green" title="E49">A199</font></span></a>, <a class="txt" href="#E8:35_1_1_5"><span class="lab"><font color="Green" title="E50">A200</font></span></a>, <a class="txt" href="#E9:35_1_1_5"><span class="lab"><font color="Green" title="E51">A201</font></span></a>, <a class="txt" href="#E10:35_1_1_5"><span class="lab"><font color="Green" title="E52">A202</font></span></a>, <a class="txt" href="#E5:35_1_1_5"><span class="lab"><font color="Green" title="E47">A197</font></span></a>, <a class="txt" href="#E6:35_1_1_5"><span class="lab"><font color="Green" title="E48">A198</font></span></a>, <a class="txt" href="#E1:35_1_1_5_2"><span class="lab"><font color="Green" title="E61">A213</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>;<br/><a NAME="E5:35_1_1_5_2_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T234" title="XREAL_1:th.234">XREAL_1:234</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_5_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:35_1_1_5"><span class="lab"><font color="Green" title="E58">A210</font></span></a>, <a class="ref" href="jordan8.html#T20" title="JORDAN8:th.20">JORDAN8:20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_5_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_5"><span class="lab"><font color="Green" title="E59">A211</font></span></a>, <a class="txt" href="#E11:35_1_1_5"><span class="lab"><font color="Green" title="E53">A203</font></span></a>, <a class="txt" href="#E19:35_1_1_5"><span class="lab"><font color="Green" title="E60">A212</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><a NAME="E21:35_1_1_5"/><span class="lab"><font color="Green" title="E61">A214</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_5_3"/><span class="lab"><font color="Green" title="E62">A215</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_5_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_5_3_1"/>
<font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_5_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A216</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:35_1_1_5_3_1"/>
<font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E14:35_1_1_5"><span class="lab"><font color="Green" title="E56">A206</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E4:35_1_1_5_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_5_3_1"><span class="lab"><font color="Green" title="E63">A216</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:35_1_1_5_3_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_5"><span class="lab"><font color="Green" title="E44">A194</font></span></a>, <a class="txt" href="#E7:35_1_1_5"><span class="lab"><font color="Green" title="E49">A199</font></span></a>, <a class="txt" href="#E8:35_1_1_5"><span class="lab"><font color="Green" title="E50">A200</font></span></a>, <a class="txt" href="#E9:35_1_1_5"><span class="lab"><font color="Green" title="E51">A201</font></span></a>, <a class="txt" href="#E10:35_1_1_5"><span class="lab"><font color="Green" title="E52">A202</font></span></a>, <a class="txt" href="#E5:35_1_1_5"><span class="lab"><font color="Green" title="E47">A197</font></span></a>, <a class="txt" href="#E6:35_1_1_5"><span class="lab"><font color="Green" title="E48">A198</font></span></a>, <a class="txt" href="#E1:35_1_1_5_3"><span class="lab"><font color="Green" title="E62">A215</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_5_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:35_1_1_5"><span class="lab"><font color="Green" title="E53">A203</font></span></a>, <a class="ref" href="jordan8.html#T18" title="JORDAN8:th.18">JORDAN8:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_5_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E18:35_1_1_5"><span class="lab"><font color="Green" title="E59">A211</font></span></a>, <a class="txt" href="#E11:35_1_1_5"><span class="lab"><font color="Green" title="E53">A203</font></span></a>, <a class="txt" href="#E21:35_1_1_5"><span class="lab"><font color="Green" title="E61">A214</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><a NAME="E23:35_1_1_5"/><span class="lab"><font color="Green" title="E62">A217</font></span>: 
<font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:35_1_1_5"><span class="lab"><font color="Green" title="E53">A203</font></span></a>, <a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>;<br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_5_4"/><span class="lab"><font color="Green" title="E63">A218</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_5_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_5_4_1"/>
<font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_5_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="nat_d.html#T36" title="NAT_D:th.36">NAT_D:36</a></span>;<br/><a NAME="E3:35_1_1_5_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_5"><span class="lab"><font color="Green" title="E59">A211</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:35_1_1_5_4_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_5"><span class="lab"><font color="Green" title="E44">A194</font></span></a>, <a class="txt" href="#E7:35_1_1_5"><span class="lab"><font color="Green" title="E49">A199</font></span></a>, <a class="txt" href="#E8:35_1_1_5"><span class="lab"><font color="Green" title="E50">A200</font></span></a>, <a class="txt" href="#E9:35_1_1_5"><span class="lab"><font color="Green" title="E51">A201</font></span></a>, <a class="txt" href="#E10:35_1_1_5"><span class="lab"><font color="Green" title="E52">A202</font></span></a>, <a class="txt" href="#E5:35_1_1_5"><span class="lab"><font color="Green" title="E47">A197</font></span></a>, <a class="txt" href="#E6:35_1_1_5"><span class="lab"><font color="Green" title="E48">A198</font></span></a>, <a class="txt" href="#E1:35_1_1_5_4"><span class="lab"><font color="Green" title="E63">A218</font></span></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></span>;<br/><a NAME="E5:35_1_1_5_4_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c14">j2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T234" title="XREAL_1:th.234">XREAL_1:234</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_5_4_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E14:35_1_1_5"><span class="lab"><font color="Green" title="E56">A206</font></span></a>, <a class="ref" href="jordan8.html#T21" title="JORDAN8:th.21">JORDAN8:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_5_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:35_1_1_5"><span class="lab"><font color="Green" title="E55">A205</font></span></a>, <a class="txt" href="#E14:35_1_1_5"><span class="lab"><font color="Green" title="E56">A206</font></span></a>, <a class="txt" href="#E23:35_1_1_5"><span class="lab"><font color="Green" title="E62">A217</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_5_5"/><span class="lab"><font color="Green" title="E63">A219</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_5_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_5_5_1"/>
<font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_5_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A220</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:35_1_1_5_5_1"/>
<font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E14:35_1_1_5"><span class="lab"><font color="Green" title="E56">A206</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E4:35_1_1_5_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_5_5_1"><span class="lab"><font color="Green" title="E64">A220</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:35_1_1_5_5_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_5"><span class="lab"><font color="Green" title="E44">A194</font></span></a>, <a class="txt" href="#E7:35_1_1_5"><span class="lab"><font color="Green" title="E49">A199</font></span></a>, <a class="txt" href="#E8:35_1_1_5"><span class="lab"><font color="Green" title="E50">A200</font></span></a>, <a class="txt" href="#E9:35_1_1_5"><span class="lab"><font color="Green" title="E51">A201</font></span></a>, <a class="txt" href="#E10:35_1_1_5"><span class="lab"><font color="Green" title="E52">A202</font></span></a>, <a class="txt" href="#E5:35_1_1_5"><span class="lab"><font color="Green" title="E47">A197</font></span></a>, <a class="txt" href="#E1:35_1_1_5_5"><span class="lab"><font color="Green" title="E63">A219</font></span></a>, <a class="ref" href="gobrd13.html#T36" title="GOBRD13:th.36">GOBRD13:36</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_5_5_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:35_1_1_5"><span class="lab"><font color="Green" title="E58">A210</font></span></a>, <a class="ref" href="jordan8.html#T18" title="JORDAN8:th.18">JORDAN8:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_5_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E18:35_1_1_5"><span class="lab"><font color="Green" title="E59">A211</font></span></a>, <a class="txt" href="#E11:35_1_1_5"><span class="lab"><font color="Green" title="E53">A203</font></span></a>, <a class="txt" href="#E21:35_1_1_5"><span class="lab"><font color="Green" title="E61">A214</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_5_6"/><span class="lab"><font color="Green" title="E63">A221</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_5_6_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_5_6_1"/>
<font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_5_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A222</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:35_1_1_5_6_1"/>
<font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:35_1_1_5"><span class="lab"><font color="Green" title="E53">A203</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E4:35_1_1_5_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_5_6_1"><span class="lab"><font color="Green" title="E64">A222</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:35_1_1_5_6_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_5"><span class="lab"><font color="Green" title="E44">A194</font></span></a>, <a class="txt" href="#E7:35_1_1_5"><span class="lab"><font color="Green" title="E49">A199</font></span></a>, <a class="txt" href="#E8:35_1_1_5"><span class="lab"><font color="Green" title="E50">A200</font></span></a>, <a class="txt" href="#E9:35_1_1_5"><span class="lab"><font color="Green" title="E51">A201</font></span></a>, <a class="txt" href="#E10:35_1_1_5"><span class="lab"><font color="Green" title="E52">A202</font></span></a>, <a class="txt" href="#E5:35_1_1_5"><span class="lab"><font color="Green" title="E47">A197</font></span></a>, <a class="txt" href="#E1:35_1_1_5_6"><span class="lab"><font color="Green" title="E63">A221</font></span></a>, <a class="ref" href="gobrd13.html#T38" title="GOBRD13:th.38">GOBRD13:38</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_5_6_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E15:35_1_1_5"><span class="lab"><font color="Green" title="E57">A207</font></span></a>, <a class="ref" href="jordan8.html#T19" title="JORDAN8:th.19">JORDAN8:19</a>, <a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_5_6"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:35_1_1_5"><span class="lab"><font color="Green" title="E55">A205</font></span></a>, <a class="txt" href="#E14:35_1_1_5"><span class="lab"><font color="Green" title="E56">A206</font></span></a>, <a class="txt" href="#E12:35_1_1_5"><span class="lab"><font color="Green" title="E54">A204</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_5_7"/><span class="lab"><font color="Green" title="E63">A223</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_5_7_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_5_7_1"/>
<font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_5_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="nat_d.html#T36" title="NAT_D:th.36">NAT_D:36</a></span>;<br/><a NAME="E3:35_1_1_5_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_5"><span class="lab"><font color="Green" title="E59">A211</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:35_1_1_5_7_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_5"><span class="lab"><font color="Green" title="E44">A194</font></span></a>, <a class="txt" href="#E7:35_1_1_5"><span class="lab"><font color="Green" title="E49">A199</font></span></a>, <a class="txt" href="#E8:35_1_1_5"><span class="lab"><font color="Green" title="E50">A200</font></span></a>, <a class="txt" href="#E9:35_1_1_5"><span class="lab"><font color="Green" title="E51">A201</font></span></a>, <a class="txt" href="#E10:35_1_1_5"><span class="lab"><font color="Green" title="E52">A202</font></span></a>, <a class="txt" href="#E5:35_1_1_5"><span class="lab"><font color="Green" title="E47">A197</font></span></a>, <a class="txt" href="#E1:35_1_1_5_7"><span class="lab"><font color="Green" title="E63">A223</font></span></a>, <a class="ref" href="gobrd13.html#T40" title="GOBRD13:th.40">GOBRD13:40</a></span>;<br/><a NAME="E5:35_1_1_5_7_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c12">j1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T234" title="XREAL_1:th.234">XREAL_1:234</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_5_7_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E15:35_1_1_5"><span class="lab"><font color="Green" title="E57">A207</font></span></a>, <a class="ref" href="jordan8.html#T21" title="JORDAN8:th.21">JORDAN8:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_5_7"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:35_1_1_5"><span class="lab"><font color="Green" title="E55">A205</font></span></a>, <a class="txt" href="#E14:35_1_1_5"><span class="lab"><font color="Green" title="E56">A206</font></span></a>, <a class="txt" href="#E23:35_1_1_5"><span class="lab"><font color="Green" title="E62">A217</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_5_8"/><span class="lab"><font color="Green" title="E63">A224</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_5_8_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_5_8_1"/>
<font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_5_8_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="nat_d.html#T36" title="NAT_D:th.36">NAT_D:36</a></span>;<br/><a NAME="E3:35_1_1_5_8_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:35_1_1_5"><span class="lab"><font color="Green" title="E55">A205</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:35_1_1_5_8_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_5"><span class="lab"><font color="Green" title="E44">A194</font></span></a>, <a class="txt" href="#E7:35_1_1_5"><span class="lab"><font color="Green" title="E49">A199</font></span></a>, <a class="txt" href="#E8:35_1_1_5"><span class="lab"><font color="Green" title="E50">A200</font></span></a>, <a class="txt" href="#E9:35_1_1_5"><span class="lab"><font color="Green" title="E51">A201</font></span></a>, <a class="txt" href="#E10:35_1_1_5"><span class="lab"><font color="Green" title="E52">A202</font></span></a>, <a class="txt" href="#E5:35_1_1_5"><span class="lab"><font color="Green" title="E47">A197</font></span></a>, <a class="txt" href="#E1:35_1_1_5_8"><span class="lab"><font color="Green" title="E63">A224</font></span></a>, <a class="ref" href="gobrd13.html#T42" title="GOBRD13:th.42">GOBRD13:42</a></span>;<br/><a NAME="E5:35_1_1_5_8_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T234" title="XREAL_1:th.234">XREAL_1:234</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_5_8_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:35_1_1_5"><span class="lab"><font color="Green" title="E58">A210</font></span></a>, <a class="ref" href="jordan8.html#T20" title="JORDAN8:th.20">JORDAN8:20</a>, <a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_5_8"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_5"><span class="lab"><font color="Green" title="E59">A211</font></span></a>, <a class="txt" href="#E11:35_1_1_5"><span class="lab"><font color="Green" title="E53">A203</font></span></a>, <a class="txt" href="#E19:35_1_1_5"><span class="lab"><font color="Green" title="E60">A212</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_5_9"/><span class="lab"><font color="Green" title="E63">A225</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_5_9_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_5_9_1"/>
<font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_5_9_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="nat_d.html#T36" title="NAT_D:th.36">NAT_D:36</a></span>;<br/><a NAME="E3:35_1_1_5_9_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_5"><span class="lab"><font color="Green" title="E59">A211</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:35_1_1_5_9_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_5"><span class="lab"><font color="Green" title="E44">A194</font></span></a>, <a class="txt" href="#E7:35_1_1_5"><span class="lab"><font color="Green" title="E49">A199</font></span></a>, <a class="txt" href="#E8:35_1_1_5"><span class="lab"><font color="Green" title="E50">A200</font></span></a>, <a class="txt" href="#E9:35_1_1_5"><span class="lab"><font color="Green" title="E51">A201</font></span></a>, <a class="txt" href="#E10:35_1_1_5"><span class="lab"><font color="Green" title="E52">A202</font></span></a>, <a class="txt" href="#E5:35_1_1_5"><span class="lab"><font color="Green" title="E47">A197</font></span></a>, <a class="txt" href="#E1:35_1_1_5_9"><span class="lab"><font color="Green" title="E63">A225</font></span></a>, <a class="ref" href="gobrd13.html#T35" title="GOBRD13:th.35">GOBRD13:35</a></span>;<br/><a NAME="E5:35_1_1_5_9_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c14">j2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T234" title="XREAL_1:th.234">XREAL_1:234</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_5_9_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E14:35_1_1_5"><span class="lab"><font color="Green" title="E56">A206</font></span></a>, <a class="ref" href="jordan8.html#T21" title="JORDAN8:th.21">JORDAN8:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_5_9"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:35_1_1_5"><span class="lab"><font color="Green" title="E55">A205</font></span></a>, <a class="txt" href="#E14:35_1_1_5"><span class="lab"><font color="Green" title="E56">A206</font></span></a>, <a class="txt" href="#E23:35_1_1_5"><span class="lab"><font color="Green" title="E62">A217</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_5_10"/><span class="lab"><font color="Green" title="E63">A226</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_5_10_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_5_10_1"/>
<font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_5_10_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A227</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:35_1_1_5_10_1"/>
<font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E14:35_1_1_5"><span class="lab"><font color="Green" title="E56">A206</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E4:35_1_1_5_10_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_5_10_1"><span class="lab"><font color="Green" title="E64">A227</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:35_1_1_5_10_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_5"><span class="lab"><font color="Green" title="E44">A194</font></span></a>, <a class="txt" href="#E7:35_1_1_5"><span class="lab"><font color="Green" title="E49">A199</font></span></a>, <a class="txt" href="#E8:35_1_1_5"><span class="lab"><font color="Green" title="E50">A200</font></span></a>, <a class="txt" href="#E9:35_1_1_5"><span class="lab"><font color="Green" title="E51">A201</font></span></a>, <a class="txt" href="#E10:35_1_1_5"><span class="lab"><font color="Green" title="E52">A202</font></span></a>, <a class="txt" href="#E5:35_1_1_5"><span class="lab"><font color="Green" title="E47">A197</font></span></a>, <a class="txt" href="#E1:35_1_1_5_10"><span class="lab"><font color="Green" title="E63">A226</font></span></a>, <a class="ref" href="gobrd13.html#T37" title="GOBRD13:th.37">GOBRD13:37</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_5_10_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:35_1_1_5"><span class="lab"><font color="Green" title="E53">A203</font></span></a>, <a class="ref" href="jordan8.html#T18" title="JORDAN8:th.18">JORDAN8:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_5_10"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E18:35_1_1_5"><span class="lab"><font color="Green" title="E59">A211</font></span></a>, <a class="txt" href="#E11:35_1_1_5"><span class="lab"><font color="Green" title="E53">A203</font></span></a>, <a class="txt" href="#E21:35_1_1_5"><span class="lab"><font color="Green" title="E61">A214</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_5_11"/><span class="lab"><font color="Green" title="E63">A228</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_5_11_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_5_11_1"/>
<font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_5_11_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="nat_d.html#T36" title="NAT_D:th.36">NAT_D:36</a></span>;<br/><a NAME="E3:35_1_1_5_11_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:35_1_1_5"><span class="lab"><font color="Green" title="E55">A205</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:35_1_1_5_11_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_5"><span class="lab"><font color="Green" title="E44">A194</font></span></a>, <a class="txt" href="#E7:35_1_1_5"><span class="lab"><font color="Green" title="E49">A199</font></span></a>, <a class="txt" href="#E8:35_1_1_5"><span class="lab"><font color="Green" title="E50">A200</font></span></a>, <a class="txt" href="#E9:35_1_1_5"><span class="lab"><font color="Green" title="E51">A201</font></span></a>, <a class="txt" href="#E10:35_1_1_5"><span class="lab"><font color="Green" title="E52">A202</font></span></a>, <a class="txt" href="#E5:35_1_1_5"><span class="lab"><font color="Green" title="E47">A197</font></span></a>, <a class="txt" href="#E1:35_1_1_5_11"><span class="lab"><font color="Green" title="E63">A228</font></span></a>, <a class="ref" href="gobrd13.html#T39" title="GOBRD13:th.39">GOBRD13:39</a></span>;<br/><a NAME="E5:35_1_1_5_11_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T234" title="XREAL_1:th.234">XREAL_1:234</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_5_11_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:35_1_1_5"><span class="lab"><font color="Green" title="E53">A203</font></span></a>, <a class="ref" href="jordan8.html#T20" title="JORDAN8:th.20">JORDAN8:20</a>, <a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_5_11"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_5"><span class="lab"><font color="Green" title="E59">A211</font></span></a>, <a class="txt" href="#E11:35_1_1_5"><span class="lab"><font color="Green" title="E53">A203</font></span></a>, <a class="txt" href="#E19:35_1_1_5"><span class="lab"><font color="Green" title="E60">A212</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_5_12"/><span class="lab"><font color="Green" title="E63">A229</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_5_12_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_5_12_1"/>
<font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_5_12_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A230</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:35_1_1_5_12_1"/>
<font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:35_1_1_5"><span class="lab"><font color="Green" title="E53">A203</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E4:35_1_1_5_12_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_5_12_1"><span class="lab"><font color="Green" title="E64">A230</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:35_1_1_5_12_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_5"><span class="lab"><font color="Green" title="E44">A194</font></span></a>, <a class="txt" href="#E7:35_1_1_5"><span class="lab"><font color="Green" title="E49">A199</font></span></a>, <a class="txt" href="#E8:35_1_1_5"><span class="lab"><font color="Green" title="E50">A200</font></span></a>, <a class="txt" href="#E9:35_1_1_5"><span class="lab"><font color="Green" title="E51">A201</font></span></a>, <a class="txt" href="#E10:35_1_1_5"><span class="lab"><font color="Green" title="E52">A202</font></span></a>, <a class="txt" href="#E5:35_1_1_5"><span class="lab"><font color="Green" title="E47">A197</font></span></a>, <a class="txt" href="#E1:35_1_1_5_12"><span class="lab"><font color="Green" title="E63">A229</font></span></a>, <a class="ref" href="gobrd13.html#T41" title="GOBRD13:th.41">GOBRD13:41</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_5_12_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E14:35_1_1_5"><span class="lab"><font color="Green" title="E56">A206</font></span></a>, <a class="ref" href="jordan8.html#T19" title="JORDAN8:th.19">JORDAN8:19</a>, <a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_5_12"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:35_1_1_5"><span class="lab"><font color="Green" title="E55">A205</font></span></a>, <a class="txt" href="#E14:35_1_1_5"><span class="lab"><font color="Green" title="E56">A206</font></span></a>, <a class="txt" href="#E12:35_1_1_5"><span class="lab"><font color="Green" title="E54">A204</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>( <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> $1 <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; ( for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> $1</span>)</span> holds <br/>(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> $1</span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> $1</span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) ) );<br/>
<a NAME="E17:35_1_1"/><span class="lab"><font color="Green" title="E44">A231</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan8.html#D1" title="JORDAN8:def.1">JORDAN8:def 1</a></span>;<br/>
<a NAME="E18:35_1_1"/><span class="lab"><font color="Green" title="E45">A232</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_1_1_6"/><span class="lab"><font color="Green" title="E46">A233</font></span>: 
2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T102" title="NEWTON:th.102">NEWTON:102</a></span>;<br/>
<a NAME="E2:35_1_1_6"/><span class="lab"><font color="Green" title="E47">A234</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:35_1_1"><span class="lab"><font color="Green" title="E44">A231</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c10">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c10">k</font>] implies <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:35_1_1_6"/><span class="lab"><font color="Green" title="E48">A235</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_6"/><span class="lab"><font color="Green" title="E49">A236</font></span>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> holds <br/>(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E5:35_1_1_6"/><span class="lab"><font color="Green" title="E50">A237</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a></span>;<br/>
<a NAME="E6:35_1_1_6"/><span class="lab"><font color="Green" title="E51">A238</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_6_1"/>
( <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_6_1_1"/><span class="lab"><font color="Green" title="E52">A239</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:35_1_1_6_1_1"/><span class="lab"><font color="Green" title="E53">A240</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_6_1_1"/><span class="lab"><font color="Green" title="E54">A241</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:35_1_1_6_1_1"/>
 <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:35_1_1_6_1_1"/>
 <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:35_1_1_6_1_1"/><span class="lab"><font color="Green" title="E55">A242</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a></span>;<br/><a NAME="E8:35_1_1_6_1_1"/>
<font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_6_1_1"><span class="lab"><font color="Green" title="E54">A241</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E9:35_1_1_6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A243</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E2:35_1_1_6"><span class="lab"><font color="Green" title="E47">A234</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_1"><span class="lab"><font color="Green" title="E53">A240</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>;<br/><div><span class="lab"><font color="Green" title="E57">A244</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">l</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:35_1_1_6_1_1_1"/>
<font color="Maroon" title="c12">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> )</span><br/><a NAME="E2:35_1_1_6_1_1_1"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">l</font> &amp; <font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_1"><span class="lab"><font color="Green" title="E52">A239</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E3:35_1_1_6_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_6_1_1_1"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_1"><span class="lab"><font color="Green" title="E55">A242</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_1"><span class="lab"><font color="Green" title="E56">A243</font></span></a>, <a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">l</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c12">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> implies for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">j1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">i2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_1_6_1_1_2"/><span class="lab"><font color="Green" title="E58">A245</font></span>: 
<font color="Maroon" title="c12">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_6_1_1_2"/><span class="lab"><font color="Green" title="E59">A246</font></span>: 
<font color="Maroon" title="c12">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">j1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">i2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E3:35_1_1_6_1_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">l</font> &amp; <font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_1"><span class="lab"><font color="Green" title="E52">A239</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_1_2"><span class="lab"><font color="Green" title="E58">A245</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E4:35_1_1_6_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:35_1_1_6_1_1_2"/>
for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">j1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">i2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_1"><span class="lab"><font color="Green" title="E52">A239</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_1_2"><span class="lab"><font color="Green" title="E59">A246</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E12:35_1_1_6_1_1"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:35_1_1_6_1_1"><span class="lab"><font color="Green" title="E57">A244</font></span></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> holds <br/>(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c12">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> &amp; <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> implies (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E13:35_1_1_6_1_1"/><span class="lab"><font color="Green" title="E58">A247</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font>
 <span class="kw">and </span><br/><a NAME="E14:35_1_1_6_1_1"/><span class="lab"><font color="Green" title="E59">A248</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><a NAME="E15:35_1_1_6_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E16:35_1_1_6_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_1"><span class="lab"><font color="Green" title="E52">A239</font></span></a>, <a class="txt" href="#E14:35_1_1_6_1_1"><span class="lab"><font color="Green" title="E59">A248</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E17:35_1_1_6_1_1"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:35_1_1_6_1_1"><span class="lab"><font color="Green" title="E58">A247</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_6_1_2"/><span class="lab"><font color="Green" title="E52">A249</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:35_1_1_6_1_2"/><span class="lab"><font color="Green" title="E53">A250</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_6_1_2"/><span class="lab"><font color="Green" title="E54">A251</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:35_1_1_6_1_2"/><span class="lab"><font color="Green" title="E55">A252</font></span>: 
 <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:35_1_1_6_1_2"/>
 <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:35_1_1_6_1_2"/><span class="lab"><font color="Green" title="E56">A253</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a></span>;<br/><a NAME="E8:35_1_1_6_1_2"/><span class="lab"><font color="Green" title="E57">A254</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E54">A251</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E9:35_1_1_6_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A255</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E2:35_1_1_6"><span class="lab"><font color="Green" title="E47">A234</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E53">A250</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>;<br/><a NAME="E10:35_1_1_6_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E53">A250</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E11:35_1_1_6_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A256</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E2:35_1_1_6"><span class="lab"><font color="Green" title="E47">A234</font></span></a>, <a class="txt" href="#E4:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E54">A251</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>;<br/><a NAME="E12:35_1_1_6_1_2"/><span class="lab"><font color="Green" title="E60">A257</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E56">A253</font></span></a>, <a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></span>;<br/><div><span class="lab"><font color="Green" title="E61">A258</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">l</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c12">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> implies for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">j1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">i2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_1_6_1_2_1"/><span class="lab"><font color="Green" title="E62">A259</font></span>: 
<font color="Maroon" title="c12">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_6_1_2_1"/><span class="lab"><font color="Green" title="E63">A260</font></span>: 
<font color="Maroon" title="c12">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">j1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">i2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E3:35_1_1_6_1_2_1"/>
<font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E52">A249</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_2_1"><span class="lab"><font color="Green" title="E62">A259</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E4:35_1_1_6_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A261</font></span>: 
( <font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T27" title="NAT_1:th.27">NAT_1:27</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c13">i1</font>, <font color="Maroon" title="c14">j1</font>, <font color="Maroon" title="c15">i2</font>, <font color="Maroon" title="c16">j2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i1</font>,<font color="Maroon" title="c14">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i2</font>,<font color="Maroon" title="c16">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i1</font>,<font color="Maroon" title="c14">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i2</font>,<font color="Maroon" title="c16">j2</font> implies <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">i2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><a NAME="E5:35_1_1_6_1_2_1"/><span class="lab"><font color="Green" title="E65">A262</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i1</font>,<font color="Maroon" title="c14">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i2</font>,<font color="Maroon" title="c16">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i1</font>,<font color="Maroon" title="c14">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i2</font>,<font color="Maroon" title="c16">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">i2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E6:35_1_1_6_1_2_1"/>
( <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E52">A249</font></span></a>, <a class="txt" href="#E12:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E60">A257</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E58">A255</font></span></a>, <a class="txt" href="#E11:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E59">A256</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_2_1"><span class="lab"><font color="Green" title="E62">A259</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_2_1"><span class="lab"><font color="Green" title="E63">A260</font></span></a>, <a class="txt" href="#E4:35_1_1_6_1_2_1"><span class="lab"><font color="Green" title="E64">A261</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_2_1"><span class="lab"><font color="Green" title="E65">A262</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E7:35_1_1_6_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A263</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">j2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E8:35_1_1_6_1_2_1"/>
( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c15">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E52">A249</font></span></a>, <a class="txt" href="#E12:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E60">A257</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E58">A255</font></span></a>, <a class="txt" href="#E11:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E59">A256</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_2_1"><span class="lab"><font color="Green" title="E62">A259</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_2_1"><span class="lab"><font color="Green" title="E63">A260</font></span></a>, <a class="txt" href="#E4:35_1_1_6_1_2_1"><span class="lab"><font color="Green" title="E64">A261</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_2_1"><span class="lab"><font color="Green" title="E65">A262</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E9:35_1_1_6_1_2_1"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">i1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E10:35_1_1_6_1_2_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">i2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_6_1_2_1"><span class="lab"><font color="Green" title="E66">A263</font></span></a>, <a class="ref" href="uniform1.html#T13" title="UNIFORM1:th.13">UNIFORM1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">l</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:35_1_1_6_1_2_2"/><span class="lab"><font color="Green" title="E62">A264</font></span>: 
<font color="Maroon" title="c12">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> )</span><br/><a NAME="E2:35_1_1_6_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E52">A249</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E3:35_1_1_6_1_2_2"/><span class="kw">then </span>
( <font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T27" title="NAT_1:th.27">NAT_1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_6_1_2_2"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E12:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E60">A257</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E58">A255</font></span></a>, <a class="txt" href="#E11:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E59">A256</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_2_2"><span class="lab"><font color="Green" title="E62">A264</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E15:35_1_1_6_1_2"/><span class="lab"><font color="Green" title="E62">A265</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E61">A258</font></span></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> holds <br/>(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><a NAME="E16:35_1_1_6_1_2"/><span class="lab"><font color="Green" title="E63">A266</font></span>: 
( <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c12">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> &amp; <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> implies (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E17:35_1_1_6_1_2"/><span class="lab"><font color="Green" title="E64">A267</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font>
 <span class="kw">and </span><br/><a NAME="E18:35_1_1_6_1_2"/><span class="lab"><font color="Green" title="E65">A268</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><a NAME="E19:35_1_1_6_1_2"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E64">A267</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E20:35_1_1_6_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A269</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E52">A249</font></span></a>, <a class="txt" href="#E18:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E65">A268</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E21:35_1_1_6_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A270</font></span>: 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E12:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E60">A257</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E58">A255</font></span></a>, <a class="txt" href="#E11:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E59">A256</font></span></a>, <a class="txt" href="#E15:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E62">A265</font></span></a>, <a class="txt" href="#E18:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E65">A268</font></span></a>, <a class="txt" href="#E16:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E63">A266</font></span></a>, <a class="ref" href="gobrd13.html#D3" title="GOBRD13:def.3">GOBRD13:def 3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_2_3"><span class="kw">now </span></a><div class="add"><a NAME="E1:35_1_1_6_1_2_3"/>
 <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c1">C</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan8.html#T12" title="JORDAN8:th.12">JORDAN8:12</a></span>;<br/><a NAME="E2:35_1_1_6_1_2_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><a NAME="E3:35_1_1_6_1_2_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_6"><span class="lab"><font color="Green" title="E46">A233</font></span></a>, <a class="ref" href="xreal_1.html#T141" title="XREAL_1:th.141">XREAL_1:141</a></span>;<br/><a NAME="E4:35_1_1_6_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A271</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E5:35_1_1_6_1_2_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E2:35_1_1_6"><span class="lab"><font color="Green" title="E47">A234</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>;<br/><a NAME="E6:35_1_1_6_1_2_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p5">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p5">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan8.html#D1" title="JORDAN8:def.1">JORDAN8:def 1</a></span>;<br/><a NAME="E7:35_1_1_6_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A272</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>;<br/><a NAME="E8:35_1_1_6_1_2_3"/><span class="lab"><font color="Green" title="E70">A273</font></span>: 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">s</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E53">A250</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E57">A254</font></span></a>, <a class="ref" href="gobrd11.html#T31" title="GOBRD11:th.31">GOBRD11:31</a></span>;<br/><span class="kw">assume </span><a NAME="E9:35_1_1_6_1_2_3"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E11:35_1_1_6_1_2_3"/><span class="lab"><font color="Green" title="E71">A274</font></span>: 
<font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E12:35_1_1_6_1_2_3"/><span class="lab"><font color="Green" title="E72">A275</font></span>: 
<font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E67">A270</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">p</font> = <font color="Maroon" title="c13">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:35_1_1_6_1_2_3"><span class="lab"><font color="Green" title="E71">A274</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">p</font> = <font color="Maroon" title="c14">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E15:35_1_1_6_1_2_3"/><span class="lab"><font color="Green" title="E73">A276</font></span>: 
<font color="Maroon" title="c15">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:35_1_1_6_1_2_3"><span class="lab"><font color="Green" title="E72">A275</font></span></a>, <a class="ref" href="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c16">r</font>, <font color="Maroon" title="c17">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E17:35_1_1_6_1_2_3"/><span class="lab"><font color="Green" title="E74">A277</font></span>: 
<font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c16">r</font>,<font color="Maroon" title="c17">s</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">and </span><br/><a NAME="E18:35_1_1_6_1_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">r</font>
 <span class="kw">and </span><br/><a NAME="E19:35_1_1_6_1_2_3"/>
<font color="Maroon" title="c16">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E20:35_1_1_6_1_2_3"/><span class="lab"><font color="Green" title="E75">A278</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:35_1_1_6_1_2_3"><span class="lab"><font color="Green" title="E71">A274</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_2_3"><span class="lab"><font color="Green" title="E70">A273</font></span></a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:35_1_1"><span class="lab"><font color="Green" title="E44">A231</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_6"><span class="lab"><font color="Green" title="E46">A233</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>
;<br/><a NAME="E22:35_1_1_6_1_2_3"/><span class="kw">then </span>
 <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c1">C</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1_6_1_2_3"><span class="lab"><font color="Green" title="E75">A278</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_2_3"><span class="lab"><font color="Green" title="E69">A272</font></span></a>, <a class="txt" href="#E4:35_1_1_6_1_2_3"><span class="lab"><font color="Green" title="E68">A271</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E23:35_1_1_6_1_2_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:35_1_1_6_1_2_3"><span class="lab"><font color="Green" title="E74">A277</font></span></a>, <a class="txt" href="#E15:35_1_1_6_1_2_3"><span class="lab"><font color="Green" title="E73">A276</font></span></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E23:35_1_1_6_1_2"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font></span><br/><a NAME="E24:35_1_1_6_1_2"/>
(  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E55">A252</font></span></a>, <a class="txt" href="#E12:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E60">A257</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E58">A255</font></span></a>, <a class="txt" href="#E11:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E59">A256</font></span></a>, <a class="txt" href="#E15:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E62">A265</font></span></a>, <a class="txt" href="#E18:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E65">A268</font></span></a>, <a class="txt" href="#E20:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E66">A269</font></span></a>, <a class="txt" href="#E16:35_1_1_6_1_2"><span class="lab"><font color="Green" title="E63">A266</font></span></a>, <a class="ref" href="gobrd13.html#D2" title="GOBRD13:def.2">GOBRD13:def 2</a>, <a class="ref" href="sprect_1.html#T13" title="SPRECT_1:th.13">SPRECT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E25:35_1_1_6_1_2"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_6_1_3"/><span class="lab"><font color="Green" title="E52">A279</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:35_1_1_6_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A280</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E3:35_1_1_6_1_3"/><span class="lab"><font color="Green" title="E54">A281</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/><a NAME="E4:35_1_1_6_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A282</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E5:35_1_1_6_1_3"/><span class="lab"><font color="Green" title="E56">A283</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">i1</font>, <font color="Maroon" title="c12">j1</font>, <font color="Maroon" title="c13">i2</font>, <font color="Maroon" title="c14">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:35_1_1_6_1_3"/><span class="lab"><font color="Green" title="E57">A284</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i1</font>,<font color="Maroon" title="c12">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:35_1_1_6_1_3"/><span class="lab"><font color="Green" title="E58">A285</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i1</font>,<font color="Maroon" title="c12">j1</font>
 <span class="kw">and </span><br/><a NAME="E9:35_1_1_6_1_3"/><span class="lab"><font color="Green" title="E59">A286</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:35_1_1_6_1_3"/><span class="lab"><font color="Green" title="E60">A287</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<font color="Maroon" title="c14">j2</font>
 <span class="kw">and </span><br/><a NAME="E11:35_1_1_6_1_3"/>
( ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>;<br/><a NAME="E12:35_1_1_6_1_3"/><span class="lab"><font color="Green" title="E61">A288</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_1_1"><span class="lab"><font color="Green" title="E43">A193</font></span></a>, <a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E4:35_1_1_6"><span class="lab"><font color="Green" title="E49">A236</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a></span>;<br/><a NAME="E13:35_1_1_6_1_3"/><span class="lab"><font color="Green" title="E62">A289</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_1_1"><span class="lab"><font color="Green" title="E43">A193</font></span></a>, <a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E4:35_1_1_6"><span class="lab"><font color="Green" title="E49">A236</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a></span>;<br/><a NAME="E14:35_1_1_6_1_3"/><span class="lab"><font color="Green" title="E63">A290</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_1_1"><span class="lab"><font color="Green" title="E43">A193</font></span></a>, <a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E4:35_1_1_6"><span class="lab"><font color="Green" title="E49">A236</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a></span>;<br/><a NAME="E15:35_1_1_6_1_3"/><span class="lab"><font color="Green" title="E64">A291</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_1_1"><span class="lab"><font color="Green" title="E43">A193</font></span></a>, <a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E4:35_1_1_6"><span class="lab"><font color="Green" title="E49">A236</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a></span>;<br/><a NAME="E16:35_1_1_6_1_3"/><span class="lab"><font color="Green" title="E65">A292</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_1_1"><span class="lab"><font color="Green" title="E43">A193</font></span></a>, <a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E4:35_1_1_6"><span class="lab"><font color="Green" title="E49">A236</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a></span>;<br/><a NAME="E17:35_1_1_6_1_3"/><span class="lab"><font color="Green" title="E66">A293</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_1_1"><span class="lab"><font color="Green" title="E43">A193</font></span></a>, <a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E4:35_1_1_6"><span class="lab"><font color="Green" title="E49">A236</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a></span>;<br/><a NAME="E18:35_1_1_6_1_3"/><span class="lab"><font color="Green" title="E67">A294</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_1_1"><span class="lab"><font color="Green" title="E43">A193</font></span></a>, <a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E4:35_1_1_6"><span class="lab"><font color="Green" title="E49">A236</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a></span>;<br/><a NAME="E19:35_1_1_6_1_3"/><span class="lab"><font color="Green" title="E68">A295</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E20:35_1_1_6_1_3"/><span class="lab"><font color="Green" title="E69">A296</font></span>: 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_6"><span class="lab"><font color="Green" title="E49">A236</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a></span>;<br/><a NAME="E21:35_1_1_6_1_3"/><span class="lab"><font color="Green" title="E70">A297</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_1_1"><span class="lab"><font color="Green" title="E43">A193</font></span></a>, <a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E4:35_1_1_6"><span class="lab"><font color="Green" title="E49">A236</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a></span>;<br/><a NAME="E22:35_1_1_6_1_3"/><span class="lab"><font color="Green" title="E71">A298</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_1_1"><span class="lab"><font color="Green" title="E43">A193</font></span></a>, <a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E4:35_1_1_6"><span class="lab"><font color="Green" title="E49">A236</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a></span>;<br/><a NAME="E23:35_1_1_6_1_3"/><span class="lab"><font color="Green" title="E72">A299</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_1_1"><span class="lab"><font color="Green" title="E43">A193</font></span></a>, <a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E4:35_1_1_6"><span class="lab"><font color="Green" title="E49">A236</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a></span>;<br/><a NAME="E24:35_1_1_6_1_3"/><span class="lab"><font color="Green" title="E73">A300</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_1_1"><span class="lab"><font color="Green" title="E43">A193</font></span></a>, <a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E4:35_1_1_6"><span class="lab"><font color="Green" title="E49">A236</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a></span>;<br/><a NAME="E25:35_1_1_6_1_3"/><span class="lab"><font color="Green" title="E74">A301</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_1_1"><span class="lab"><font color="Green" title="E43">A193</font></span></a>, <a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E4:35_1_1_6"><span class="lab"><font color="Green" title="E49">A236</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a></span>;<br/><a NAME="E26:35_1_1_6_1_3"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T35" title="NAT_D:th.35">NAT_D:35</a></span>;<br/><a NAME="E27:35_1_1_6_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A302</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E28:35_1_1_6_1_3"/><span class="lab"><font color="Green" title="E76">A303</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><span class="kw">thus </span><a NAME="E29:35_1_1_6_1_3"/><span class="lab"><font color="Green" title="E77">A304</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> holds <br/>(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_3_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_6_1_3_1_1"/>
( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> ) or (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_6_1_3_1_1_1"/>
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">i</font>, <font color="Maroon" title="c16">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:35_1_1_6_1_3_1_1_1"/><span class="lab"><font color="Green" title="E78">A305</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_6_1_3_1_1_1"/><span class="lab"><font color="Green" title="E79">A306</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a>, <a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E20:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E69">A296</font></span></a></span>;<br/><span class="kw">thus </span><a NAME="E5:35_1_1_6_1_3_1_1_1"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c17">f</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/>
<a NAME="E1:35_1_1_6_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E80">A307</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></span>;<br/>
<a NAME="E2:35_1_1_6_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E81">A308</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i1</font>,<font color="Maroon" title="c12">j1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<font color="Maroon" title="c14">j2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E27:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E75">A302</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E53">A280</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_6_1_3_1_1_1_1_1"/>
( ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E4:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E55">A282</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_1_1_1"><span class="lab"><font color="Green" title="E78">A305</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E81">A308</font></span></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_1_1_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:35_1_1_6_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E82">A309</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_6_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E83">A310</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">i19</font>, <font color="Maroon" title="c19">j19</font>, <font color="Maroon" title="c20">i29</font>, <font color="Maroon" title="c21">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font> implies <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><a NAME="E1:35_1_1_6_1_3_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E84">A311</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E2:35_1_1_6_1_3_1_1_1_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j19</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E12:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E61">A288</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E82">A309</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E3:35_1_1_6_1_3_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A312</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E4:35_1_1_6_1_3_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i19</font> &amp; <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E12:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E61">A288</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E82">A309</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E84">A311</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><span class="kw">hence </span><a NAME="E5:35_1_1_6_1_3_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6_1_3_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E85">A312</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:35_1_1_6_1_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E12:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E61">A288</font></span></a>, <a class="txt" href="#E4:35_1_1_6_1_3_1_1_1"><span class="lab"><font color="Green" title="E79">A306</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E80">A307</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E82">A309</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E83">A310</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_1_1_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:35_1_1_6_1_3_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E82">A313</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_6_1_3_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E83">A314</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">i19</font>, <font color="Maroon" title="c19">j19</font>, <font color="Maroon" title="c20">i29</font>, <font color="Maroon" title="c21">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font> implies <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><a NAME="E1:35_1_1_6_1_3_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E84">A315</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E2:35_1_1_6_1_3_1_1_1_1_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j19</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E18:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E67">A294</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E82">A313</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E3:35_1_1_6_1_3_1_1_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">j19</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">j29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E68">A295</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><a NAME="E4:35_1_1_6_1_3_1_1_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A316</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j19</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">j29</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E5:35_1_1_6_1_3_1_1_1_1_1_2_1"/>
( <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i19</font> &amp; <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E18:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E67">A294</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E82">A313</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E84">A315</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E6:35_1_1_6_1_3_1_1_1_1_1_2_1"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:35_1_1_6_1_3_1_1_1_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_6_1_3_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E85">A316</font></span></a>, <a class="ref" href="uniform1.html#T13" title="UNIFORM1:th.13">UNIFORM1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:35_1_1_6_1_3_1_1_1_1_1_2"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E18:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E67">A294</font></span></a>, <a class="txt" href="#E4:35_1_1_6_1_3_1_1_1"><span class="lab"><font color="Green" title="E79">A306</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E80">A307</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E82">A313</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E83">A314</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_1_1_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:35_1_1_6_1_3_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E82">A317</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_6_1_3_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E83">A318</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_1_1_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">i19</font>, <font color="Maroon" title="c19">j19</font>, <font color="Maroon" title="c20">i29</font>, <font color="Maroon" title="c21">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font> implies <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><a NAME="E1:35_1_1_6_1_3_1_1_1_1_1_3_1"/><span class="lab"><font color="Green" title="E84">A319</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E2:35_1_1_6_1_3_1_1_1_1_1_3_1"/><span class="kw">then </span>
( <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i19</font> &amp; <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E17:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E66">A293</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E82">A317</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E3:35_1_1_6_1_3_1_1_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A320</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E4:35_1_1_6_1_3_1_1_1_1_1_3_1"/>
( <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j19</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E17:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E66">A293</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E82">A317</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_1_1_1_3_1"><span class="lab"><font color="Green" title="E84">A319</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><span class="kw">hence </span><a NAME="E5:35_1_1_6_1_3_1_1_1_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6_1_3_1_1_1_1_1_3_1"><span class="lab"><font color="Green" title="E85">A320</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:35_1_1_6_1_3_1_1_1_1_1_3"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E17:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E66">A293</font></span></a>, <a class="txt" href="#E4:35_1_1_6_1_3_1_1_1"><span class="lab"><font color="Green" title="E79">A306</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E80">A307</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E82">A317</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E83">A318</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_1_1_1_4"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:35_1_1_6_1_3_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E82">A321</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_6_1_3_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E83">A322</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_1_1_1_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">i19</font>, <font color="Maroon" title="c19">j19</font>, <font color="Maroon" title="c20">i29</font>, <font color="Maroon" title="c21">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font> implies <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><a NAME="E1:35_1_1_6_1_3_1_1_1_1_1_4_1"/><span class="lab"><font color="Green" title="E84">A323</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E2:35_1_1_6_1_3_1_1_1_1_1_4_1"/><span class="kw">then </span>
( <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i19</font> &amp; <font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E24:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E73">A300</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E82">A321</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E3:35_1_1_6_1_3_1_1_1_1_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">i19</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">i29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E76">A303</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><a NAME="E4:35_1_1_6_1_3_1_1_1_1_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A324</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i19</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">i29</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E5:35_1_1_6_1_3_1_1_1_1_1_4_1"/>
( <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j19</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E24:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E73">A300</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E82">A321</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_1_1_1_4_1"><span class="lab"><font color="Green" title="E84">A323</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E6:35_1_1_6_1_3_1_1_1_1_1_4_1"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:35_1_1_6_1_3_1_1_1_1_1_4_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_6_1_3_1_1_1_1_1_4_1"><span class="lab"><font color="Green" title="E85">A324</font></span></a>, <a class="ref" href="uniform1.html#T13" title="UNIFORM1:th.13">UNIFORM1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:35_1_1_6_1_3_1_1_1_1_1_4"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E24:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E73">A300</font></span></a>, <a class="txt" href="#E4:35_1_1_6_1_3_1_1_1"><span class="lab"><font color="Green" title="E79">A306</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E80">A307</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E82">A321</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E83">A322</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_6_1_3_1_1_2"/><span class="lab"><font color="Green" title="E78">A325</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">i</font>, <font color="Maroon" title="c16">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:35_1_1_6_1_3_1_1_2"/><span class="lab"><font color="Green" title="E79">A326</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_6_1_3_1_1_2"/><span class="lab"><font color="Green" title="E80">A327</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a>, <a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E20:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E69">A296</font></span></a></span>;<br/><span class="kw">thus </span><a NAME="E5:35_1_1_6_1_3_1_1_2"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c17">f</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/>
<a NAME="E1:35_1_1_6_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E81">A328</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></span>;<br/>
<a NAME="E2:35_1_1_6_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E82">A329</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i1</font>,<font color="Maroon" title="c12">j1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<font color="Maroon" title="c14">j2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E27:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E75">A302</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E53">A280</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_6_1_3_1_1_2_1_1"/>
( ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E4:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E55">A282</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_1_1_2"><span class="lab"><font color="Green" title="E79">A326</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E82">A329</font></span></a>, <a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_2_1_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:35_1_1_6_1_3_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E83">A330</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_6_1_3_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E84">A331</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_2_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">i19</font>, <font color="Maroon" title="c19">j19</font>, <font color="Maroon" title="c20">i29</font>, <font color="Maroon" title="c21">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font> implies <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><a NAME="E1:35_1_1_6_1_3_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E85">A332</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E2:35_1_1_6_1_3_1_1_2_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i19</font> &amp; <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E23:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E72">A299</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2"><span class="lab"><font color="Green" title="E78">A325</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E83">A330</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E3:35_1_1_6_1_3_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A333</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E4:35_1_1_6_1_3_1_1_2_1_1_1_1"/>
( <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j19</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E23:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E72">A299</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2"><span class="lab"><font color="Green" title="E78">A325</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E83">A330</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E85">A332</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><span class="kw">hence </span><a NAME="E5:35_1_1_6_1_3_1_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6_1_3_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E86">A333</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:35_1_1_6_1_3_1_1_2_1_1_1"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E23:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E72">A299</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2"><span class="lab"><font color="Green" title="E78">A325</font></span></a>, <a class="txt" href="#E4:35_1_1_6_1_3_1_1_2"><span class="lab"><font color="Green" title="E80">A327</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E81">A328</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E83">A330</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E84">A331</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_2_1_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:35_1_1_6_1_3_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E83">A334</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_6_1_3_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E84">A335</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">i19</font>, <font color="Maroon" title="c19">j19</font>, <font color="Maroon" title="c20">i29</font>, <font color="Maroon" title="c21">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font> implies <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><a NAME="E1:35_1_1_6_1_3_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E85">A336</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E2:35_1_1_6_1_3_1_1_2_1_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j19</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E22:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E71">A298</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2"><span class="lab"><font color="Green" title="E78">A325</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E83">A334</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E3:35_1_1_6_1_3_1_1_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A337</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E4:35_1_1_6_1_3_1_1_2_1_1_2_1"/>
( <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i19</font> &amp; <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E22:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E71">A298</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2"><span class="lab"><font color="Green" title="E78">A325</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E83">A334</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E85">A336</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><span class="kw">hence </span><a NAME="E5:35_1_1_6_1_3_1_1_2_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6_1_3_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E86">A337</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:35_1_1_6_1_3_1_1_2_1_1_2"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E22:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E71">A298</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2"><span class="lab"><font color="Green" title="E78">A325</font></span></a>, <a class="txt" href="#E4:35_1_1_6_1_3_1_1_2"><span class="lab"><font color="Green" title="E80">A327</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E81">A328</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E83">A334</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E84">A335</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_2_1_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:35_1_1_6_1_3_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E83">A338</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_6_1_3_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E84">A339</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_2_1_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">i19</font>, <font color="Maroon" title="c19">j19</font>, <font color="Maroon" title="c20">i29</font>, <font color="Maroon" title="c21">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font> implies <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><a NAME="E1:35_1_1_6_1_3_1_1_2_1_1_3_1"/><span class="lab"><font color="Green" title="E85">A340</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E2:35_1_1_6_1_3_1_1_2_1_1_3_1"/><span class="kw">then </span>
( <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i19</font> &amp; <font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E21:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E70">A297</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2"><span class="lab"><font color="Green" title="E78">A325</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E83">A338</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E3:35_1_1_6_1_3_1_1_2_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">i19</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">i29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E76">A303</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><a NAME="E4:35_1_1_6_1_3_1_1_2_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A341</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i19</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">i29</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E5:35_1_1_6_1_3_1_1_2_1_1_3_1"/>
( <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j19</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E21:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E70">A297</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2"><span class="lab"><font color="Green" title="E78">A325</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E83">A338</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2_1_1_3_1"><span class="lab"><font color="Green" title="E85">A340</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E6:35_1_1_6_1_3_1_1_2_1_1_3_1"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:35_1_1_6_1_3_1_1_2_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_6_1_3_1_1_2_1_1_3_1"><span class="lab"><font color="Green" title="E86">A341</font></span></a>, <a class="ref" href="uniform1.html#T13" title="UNIFORM1:th.13">UNIFORM1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:35_1_1_6_1_3_1_1_2_1_1_3"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E21:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E70">A297</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2"><span class="lab"><font color="Green" title="E78">A325</font></span></a>, <a class="txt" href="#E4:35_1_1_6_1_3_1_1_2"><span class="lab"><font color="Green" title="E80">A327</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E81">A328</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E83">A338</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E84">A339</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_2_1_1_4"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:35_1_1_6_1_3_1_1_2_1_1_4"/><span class="lab"><font color="Green" title="E83">A342</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_6_1_3_1_1_2_1_1_4"/><span class="lab"><font color="Green" title="E84">A343</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_2_1_1_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">i19</font>, <font color="Maroon" title="c19">j19</font>, <font color="Maroon" title="c20">i29</font>, <font color="Maroon" title="c21">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font> implies <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><a NAME="E1:35_1_1_6_1_3_1_1_2_1_1_4_1"/><span class="lab"><font color="Green" title="E85">A344</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E2:35_1_1_6_1_3_1_1_2_1_1_4_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j19</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E16:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E65">A292</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2"><span class="lab"><font color="Green" title="E78">A325</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E83">A342</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E3:35_1_1_6_1_3_1_1_2_1_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">j19</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">j29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E68">A295</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><a NAME="E4:35_1_1_6_1_3_1_1_2_1_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A345</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j19</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">j29</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E5:35_1_1_6_1_3_1_1_2_1_1_4_1"/>
( <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i19</font> &amp; <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E16:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E65">A292</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2"><span class="lab"><font color="Green" title="E78">A325</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E83">A342</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2_1_1_4_1"><span class="lab"><font color="Green" title="E85">A344</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E6:35_1_1_6_1_3_1_1_2_1_1_4_1"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:35_1_1_6_1_3_1_1_2_1_1_4_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_6_1_3_1_1_2_1_1_4_1"><span class="lab"><font color="Green" title="E86">A345</font></span></a>, <a class="ref" href="uniform1.html#T13" title="UNIFORM1:th.13">UNIFORM1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:35_1_1_6_1_3_1_1_2_1_1_4"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E16:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E65">A292</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2"><span class="lab"><font color="Green" title="E78">A325</font></span></a>, <a class="txt" href="#E4:35_1_1_6_1_3_1_1_2"><span class="lab"><font color="Green" title="E80">A327</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E81">A328</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E83">A342</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E84">A343</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_6_1_3_1_1_3"/><span class="lab"><font color="Green" title="E78">A346</font></span>: 
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">i</font>, <font color="Maroon" title="c16">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:35_1_1_6_1_3_1_1_3"/><span class="lab"><font color="Green" title="E79">A347</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_6_1_3_1_1_3"/><span class="lab"><font color="Green" title="E80">A348</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a>, <a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E20:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E69">A296</font></span></a></span>;<br/><span class="kw">thus </span><a NAME="E5:35_1_1_6_1_3_1_1_3"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c17">f</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/>
<a NAME="E1:35_1_1_6_1_3_1_1_3_1"/><span class="lab"><font color="Green" title="E81">A349</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></span>;<br/>
<a NAME="E2:35_1_1_6_1_3_1_1_3_1"/><span class="lab"><font color="Green" title="E82">A350</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i1</font>,<font color="Maroon" title="c12">j1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<font color="Maroon" title="c14">j2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E27:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E75">A302</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E53">A280</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_6_1_3_1_1_3_1_1"/>
( ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E4:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E55">A282</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_1_1_3"><span class="lab"><font color="Green" title="E79">A347</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_1_1_3_1"><span class="lab"><font color="Green" title="E82">A350</font></span></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_3_1_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:35_1_1_6_1_3_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E83">A351</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_6_1_3_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E84">A352</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_3_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">i19</font>, <font color="Maroon" title="c19">j19</font>, <font color="Maroon" title="c20">i29</font>, <font color="Maroon" title="c21">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font> implies <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><a NAME="E1:35_1_1_6_1_3_1_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E85">A353</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E2:35_1_1_6_1_3_1_1_3_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i19</font> &amp; <font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E15:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E64">A291</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3"><span class="lab"><font color="Green" title="E78">A346</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E83">A351</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E3:35_1_1_6_1_3_1_1_3_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">i19</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">i29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E76">A303</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><a NAME="E4:35_1_1_6_1_3_1_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A354</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i19</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">i29</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E5:35_1_1_6_1_3_1_1_3_1_1_1_1"/>
( <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j19</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E15:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E64">A291</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3"><span class="lab"><font color="Green" title="E78">A346</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E83">A351</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E85">A353</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E6:35_1_1_6_1_3_1_1_3_1_1_1_1"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:35_1_1_6_1_3_1_1_3_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_6_1_3_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E86">A354</font></span></a>, <a class="ref" href="uniform1.html#T13" title="UNIFORM1:th.13">UNIFORM1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:35_1_1_6_1_3_1_1_3_1_1_1"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E15:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E64">A291</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3"><span class="lab"><font color="Green" title="E78">A346</font></span></a>, <a class="txt" href="#E4:35_1_1_6_1_3_1_1_3"><span class="lab"><font color="Green" title="E80">A348</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3_1"><span class="lab"><font color="Green" title="E81">A349</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E83">A351</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E84">A352</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_3_1_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:35_1_1_6_1_3_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E83">A355</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_6_1_3_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E84">A356</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_3_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">i19</font>, <font color="Maroon" title="c19">j19</font>, <font color="Maroon" title="c20">i29</font>, <font color="Maroon" title="c21">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font> implies <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><a NAME="E1:35_1_1_6_1_3_1_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E85">A357</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E2:35_1_1_6_1_3_1_1_3_1_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i19</font> &amp; <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E14:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E63">A290</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3"><span class="lab"><font color="Green" title="E78">A346</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E83">A355</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E3:35_1_1_6_1_3_1_1_3_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A358</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E4:35_1_1_6_1_3_1_1_3_1_1_2_1"/>
( <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j19</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E14:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E63">A290</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3"><span class="lab"><font color="Green" title="E78">A346</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E83">A355</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E85">A357</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><span class="kw">hence </span><a NAME="E5:35_1_1_6_1_3_1_1_3_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6_1_3_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E86">A358</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:35_1_1_6_1_3_1_1_3_1_1_2"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E14:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E63">A290</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3"><span class="lab"><font color="Green" title="E78">A346</font></span></a>, <a class="txt" href="#E4:35_1_1_6_1_3_1_1_3"><span class="lab"><font color="Green" title="E80">A348</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3_1"><span class="lab"><font color="Green" title="E81">A349</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E83">A355</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E84">A356</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_3_1_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:35_1_1_6_1_3_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E83">A359</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_6_1_3_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E84">A360</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_3_1_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">i19</font>, <font color="Maroon" title="c19">j19</font>, <font color="Maroon" title="c20">i29</font>, <font color="Maroon" title="c21">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font> implies <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><a NAME="E1:35_1_1_6_1_3_1_1_3_1_1_3_1"/><span class="lab"><font color="Green" title="E85">A361</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E2:35_1_1_6_1_3_1_1_3_1_1_3_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j19</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E13:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E62">A289</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3"><span class="lab"><font color="Green" title="E78">A346</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E83">A359</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E3:35_1_1_6_1_3_1_1_3_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">j19</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">j29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E68">A295</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><a NAME="E4:35_1_1_6_1_3_1_1_3_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A362</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j19</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">j29</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E5:35_1_1_6_1_3_1_1_3_1_1_3_1"/>
( <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i19</font> &amp; <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E13:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E62">A289</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3"><span class="lab"><font color="Green" title="E78">A346</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E83">A359</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3_1_1_3_1"><span class="lab"><font color="Green" title="E85">A361</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E6:35_1_1_6_1_3_1_1_3_1_1_3_1"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:35_1_1_6_1_3_1_1_3_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_6_1_3_1_1_3_1_1_3_1"><span class="lab"><font color="Green" title="E86">A362</font></span></a>, <a class="ref" href="uniform1.html#T13" title="UNIFORM1:th.13">UNIFORM1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:35_1_1_6_1_3_1_1_3_1_1_3"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E13:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E62">A289</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3"><span class="lab"><font color="Green" title="E78">A346</font></span></a>, <a class="txt" href="#E4:35_1_1_6_1_3_1_1_3"><span class="lab"><font color="Green" title="E80">A348</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3_1"><span class="lab"><font color="Green" title="E81">A349</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E83">A359</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E84">A360</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_3_1_1_4"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:35_1_1_6_1_3_1_1_3_1_1_4"/><span class="lab"><font color="Green" title="E83">A363</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_6_1_3_1_1_3_1_1_4"/><span class="lab"><font color="Green" title="E84">A364</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_3_1_1_3_1_1_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">i19</font>, <font color="Maroon" title="c19">j19</font>, <font color="Maroon" title="c20">i29</font>, <font color="Maroon" title="c21">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font> implies <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><a NAME="E1:35_1_1_6_1_3_1_1_3_1_1_4_1"/><span class="lab"><font color="Green" title="E85">A365</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i19</font>,<font color="Maroon" title="c19">j19</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i29</font>,<font color="Maroon" title="c21">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E2:35_1_1_6_1_3_1_1_3_1_1_4_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j19</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E25:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E74">A301</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3"><span class="lab"><font color="Green" title="E78">A346</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E83">A363</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E3:35_1_1_6_1_3_1_1_3_1_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A366</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E4:35_1_1_6_1_3_1_1_3_1_1_4_1"/>
( <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i19</font> &amp; <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E25:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E74">A301</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3"><span class="lab"><font color="Green" title="E78">A346</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E83">A363</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E85">A365</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><span class="kw">hence </span><a NAME="E5:35_1_1_6_1_3_1_1_3_1_1_4_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6_1_3_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E86">A366</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:35_1_1_6_1_3_1_1_3_1_1_4"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E25:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E74">A301</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3"><span class="lab"><font color="Green" title="E78">A346</font></span></a>, <a class="txt" href="#E4:35_1_1_6_1_3_1_1_3"><span class="lab"><font color="Green" title="E80">A348</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3_1"><span class="lab"><font color="Green" title="E81">A349</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E83">A363</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E84">A364</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">let </span><font color="Maroon" title="c15">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font> &amp; <font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> implies (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E30:35_1_1_6_1_3"/><span class="lab"><font color="Green" title="E78">A367</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font>
 <span class="kw">and </span><br/><a NAME="E31:35_1_1_6_1_3"/><span class="lab"><font color="Green" title="E79">A368</font></span>: 
<font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><a NAME="E32:35_1_1_6_1_3"/><span class="lab"><font color="Green" title="E80">A369</font></span>: 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_6"><span class="lab"><font color="Green" title="E49">A236</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_3_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_6_1_3_2_1"/>
( <font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> or <font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_6_1_3_2_1_1"/><span class="lab"><font color="Green" title="E81">A370</font></span>: 
<font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E2:35_1_1_6_1_3_2_1_1"/><span class="lab"><font color="Green" title="E82">A371</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E68">A295</font></span></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/><a NAME="E3:35_1_1_6_1_3_2_1_1"/><span class="lab"><font color="Green" title="E83">A372</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E76">A303</font></span></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/><span class="kw">thus </span><a NAME="E4:35_1_1_6_1_3_2_1_1"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_6_1_3_2_1_1_1_1"/>
( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> ) or (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_6_1_3_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E84">A373</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E2:35_1_1_6_1_3_2_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A374</font></span>: 
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a>, <a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E20:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E69">A296</font></span></a></span>;<br/><a NAME="E3:35_1_1_6_1_3_2_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A375</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<font color="Maroon" title="c14">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E53">A280</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/><a NAME="E4:35_1_1_6_1_3_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E87">A376</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i1</font>,<font color="Maroon" title="c12">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E27:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E75">A302</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E85">A374</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_6_1_3_2_1_1_1_1_1_1_1"/>
( ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E4:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E55">A282</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E85">A374</font></span></a>, <a class="txt" href="#E4:35_1_1_6_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E87">A376</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E86">A375</font></span></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_1_1_1_1_1_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:35_1_1_6_1_3_2_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E88">A377</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_6_1_3_2_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E89">A378</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E3:35_1_1_6_1_3_2_1_1_1_1_1_1_1_1"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i1</font>,<font color="Maroon" title="c14">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E88">A377</font></span></a>, <a class="ref" href="gobrd13.html#T36" title="GOBRD13:th.36">GOBRD13:36</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_6_1_3_2_1_1_1_1_1_1_1_1"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E12:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E61">A288</font></span></a>, <a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E81">A370</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E84">A373</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E86">A375</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E88">A377</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E89">A378</font></span></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font></span><br/><a NAME="E5:35_1_1_6_1_3_2_1_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j1</font> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i1</font>,<font color="Maroon" title="c12">j1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E20:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E69">A296</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E88">A377</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_6_1_3_2_1_1_1_1_1_1_1_1"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E12:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E61">A288</font></span></a>, <a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E81">A370</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E86">A375</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E88">A377</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E89">A378</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:35_1_1_6_1_3_2_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E88">A379</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_6_1_3_2_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E89">A380</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E3:35_1_1_6_1_3_2_1_1_1_1_1_1_1_2"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E88">A379</font></span></a>, <a class="ref" href="gobrd13.html#T38" title="GOBRD13:th.38">GOBRD13:38</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_6_1_3_2_1_1_1_1_1_1_1_2"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E18:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E67">A294</font></span></a>, <a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E81">A370</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E82">A371</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E84">A373</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E86">A375</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E88">A379</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E89">A380</font></span></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font></span><br/><a NAME="E5:35_1_1_6_1_3_2_1_1_1_1_1_1_1_2"/>
( <font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i1</font> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E20:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E69">A296</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E88">A379</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_6_1_3_2_1_1_1_1_1_1_1_2"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E18:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E67">A294</font></span></a>, <a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E81">A370</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E82">A371</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E86">A375</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E88">A379</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E89">A380</font></span></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_1_1_1_1_1_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:35_1_1_6_1_3_2_1_1_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E88">A381</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_6_1_3_2_1_1_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E89">A382</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E3:35_1_1_6_1_3_2_1_1_1_1_1_1_1_3"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E88">A381</font></span></a>, <a class="ref" href="gobrd13.html#T40" title="GOBRD13:th.40">GOBRD13:40</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_6_1_3_2_1_1_1_1_1_1_1_3"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E17:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E66">A293</font></span></a>, <a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E81">A370</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E84">A373</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E86">A375</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E88">A381</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E89">A382</font></span></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font></span><br/><a NAME="E5:35_1_1_6_1_3_2_1_1_1_1_1_1_1_3"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c13">i2</font>,<font color="Maroon" title="c14">j2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E20:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E69">A296</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E88">A381</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_6_1_3_2_1_1_1_1_1_1_1_3"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E17:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E66">A293</font></span></a>, <a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E81">A370</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E86">A375</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E88">A381</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E89">A382</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_1_1_1_1_1_1_4"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:35_1_1_6_1_3_2_1_1_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E88">A383</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_6_1_3_2_1_1_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E89">A384</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E3:35_1_1_6_1_3_2_1_1_1_1_1_1_1_4"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E88">A383</font></span></a>, <a class="ref" href="gobrd13.html#T42" title="GOBRD13:th.42">GOBRD13:42</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_6_1_3_2_1_1_1_1_1_1_1_4"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E24:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E73">A300</font></span></a>, <a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E81">A370</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E83">A372</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E84">A373</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E86">A375</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E88">A383</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E89">A384</font></span></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font></span><br/><a NAME="E5:35_1_1_6_1_3_2_1_1_1_1_1_1_1_4"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E20:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E69">A296</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E88">A383</font></span></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_6_1_3_2_1_1_1_1_1_1_1_4"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E24:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E73">A300</font></span></a>, <a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E81">A370</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E83">A372</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E86">A375</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E88">A383</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E89">A384</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:35_1_1_6_1_3_2_1_1_1_1_1"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_6_1_3_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E84">A385</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E2:35_1_1_6_1_3_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A386</font></span>: 
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a>, <a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E20:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E69">A296</font></span></a></span>;<br/><a NAME="E3:35_1_1_6_1_3_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A387</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<font color="Maroon" title="c14">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E53">A280</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/><a NAME="E4:35_1_1_6_1_3_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E87">A388</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i1</font>,<font color="Maroon" title="c12">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E27:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E75">A302</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E85">A386</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_1_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_6_1_3_2_1_1_1_1_2_1_1"/>
( ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E4:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E55">A282</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E85">A386</font></span></a>, <a class="txt" href="#E4:35_1_1_6_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E87">A388</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E86">A387</font></span></a>, <a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_1_1_1_2_1_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:35_1_1_6_1_3_2_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E88">A389</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_6_1_3_2_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E89">A390</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E3:35_1_1_6_1_3_2_1_1_1_1_2_1_1_1"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E88">A389</font></span></a>, <a class="ref" href="gobrd13.html#T35" title="GOBRD13:th.35">GOBRD13:35</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_6_1_3_2_1_1_1_1_2_1_1_1"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E23:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E72">A299</font></span></a>, <a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E81">A370</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E84">A385</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E86">A387</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E88">A389</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E89">A390</font></span></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font></span><br/><a NAME="E5:35_1_1_6_1_3_2_1_1_1_1_2_1_1_1"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i1</font>,<font color="Maroon" title="c14">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E88">A389</font></span></a>, <a class="ref" href="gobrd13.html#T36" title="GOBRD13:th.36">GOBRD13:36</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_6_1_3_2_1_1_1_1_2_1_1_1"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E23:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E72">A299</font></span></a>, <a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E81">A370</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E84">A385</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E86">A387</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E88">A389</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E89">A390</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:35_1_1_6_1_3_2_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E88">A391</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_6_1_3_2_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E89">A392</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E3:35_1_1_6_1_3_2_1_1_1_1_2_1_1_2"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c13">i2</font>,<font color="Maroon" title="c14">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E88">A391</font></span></a>, <a class="ref" href="gobrd13.html#T37" title="GOBRD13:th.37">GOBRD13:37</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_6_1_3_2_1_1_1_1_2_1_1_2"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E22:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E71">A298</font></span></a>, <a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E81">A370</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E84">A385</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E86">A387</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E88">A391</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E89">A392</font></span></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font></span><br/><a NAME="E5:35_1_1_6_1_3_2_1_1_1_1_2_1_1_2"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E88">A391</font></span></a>, <a class="ref" href="gobrd13.html#T38" title="GOBRD13:th.38">GOBRD13:38</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_6_1_3_2_1_1_1_1_2_1_1_2"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E22:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E71">A298</font></span></a>, <a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E81">A370</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E84">A385</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E86">A387</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E88">A391</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E89">A392</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_1_1_1_2_1_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:35_1_1_6_1_3_2_1_1_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E88">A393</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_6_1_3_2_1_1_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E89">A394</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E3:35_1_1_6_1_3_2_1_1_1_1_2_1_1_3"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E88">A393</font></span></a>, <a class="ref" href="gobrd13.html#T39" title="GOBRD13:th.39">GOBRD13:39</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_6_1_3_2_1_1_1_1_2_1_1_3"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E21:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E70">A297</font></span></a>, <a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E81">A370</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E83">A372</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E84">A385</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E86">A387</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E88">A393</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E89">A394</font></span></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font></span><br/><a NAME="E5:35_1_1_6_1_3_2_1_1_1_1_2_1_1_3"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E88">A393</font></span></a>, <a class="ref" href="gobrd13.html#T40" title="GOBRD13:th.40">GOBRD13:40</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_6_1_3_2_1_1_1_1_2_1_1_3"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E21:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E70">A297</font></span></a>, <a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E81">A370</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E83">A372</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E84">A385</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E86">A387</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E88">A393</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E89">A394</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_1_1_1_2_1_1_4"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:35_1_1_6_1_3_2_1_1_1_1_2_1_1_4"/><span class="lab"><font color="Green" title="E88">A395</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_6_1_3_2_1_1_1_1_2_1_1_4"/><span class="lab"><font color="Green" title="E89">A396</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E3:35_1_1_6_1_3_2_1_1_1_1_2_1_1_4"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E88">A395</font></span></a>, <a class="ref" href="gobrd13.html#T41" title="GOBRD13:th.41">GOBRD13:41</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_6_1_3_2_1_1_1_1_2_1_1_4"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E16:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E65">A292</font></span></a>, <a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E81">A370</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E82">A371</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E84">A385</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E86">A387</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E88">A395</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E89">A396</font></span></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font></span><br/><a NAME="E5:35_1_1_6_1_3_2_1_1_1_1_2_1_1_4"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E88">A395</font></span></a>, <a class="ref" href="gobrd13.html#T42" title="GOBRD13:th.42">GOBRD13:42</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_6_1_3_2_1_1_1_1_2_1_1_4"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E16:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E65">A292</font></span></a>, <a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E81">A370</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E82">A371</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E84">A385</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E86">A387</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E88">A395</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E89">A396</font></span></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:35_1_1_6_1_3_2_1_1_1_1_2"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_6_1_3_2_1_1_1_1_3"/><span class="lab"><font color="Green" title="E84">A397</font></span>: 
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E2:35_1_1_6_1_3_2_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A398</font></span>: 
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a>, <a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E20:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E69">A296</font></span></a></span>;<br/><a NAME="E3:35_1_1_6_1_3_2_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A399</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<font color="Maroon" title="c14">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E53">A280</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/><a NAME="E4:35_1_1_6_1_3_2_1_1_1_1_3"/><span class="lab"><font color="Green" title="E87">A400</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i1</font>,<font color="Maroon" title="c12">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E27:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E75">A302</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E85">A398</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_1_1_1_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_1_1_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_6_1_3_2_1_1_1_1_3_1_1"/>
( ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E4:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E55">A282</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E85">A398</font></span></a>, <a class="txt" href="#E4:35_1_1_6_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E87">A400</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E86">A399</font></span></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_1_1_1_3_1_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:35_1_1_6_1_3_2_1_1_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E88">A401</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_6_1_3_2_1_1_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E89">A402</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E3:35_1_1_6_1_3_2_1_1_1_1_3_1_1_1"/>
( <font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j1</font> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E32:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E80">A369</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E88">A401</font></span></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_6_1_3_2_1_1_1_1_3_1_1_1"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E15:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E64">A291</font></span></a>, <a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E81">A370</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E83">A372</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E84">A397</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E86">A399</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E88">A401</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E89">A402</font></span></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font></span><br/><a NAME="E5:35_1_1_6_1_3_2_1_1_1_1_3_1_1_1"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E88">A401</font></span></a>, <a class="ref" href="gobrd13.html#T35" title="GOBRD13:th.35">GOBRD13:35</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_6_1_3_2_1_1_1_1_3_1_1_1"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E15:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E64">A291</font></span></a>, <a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E81">A370</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E83">A372</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E84">A397</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E86">A399</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E88">A401</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E89">A402</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_1_1_1_3_1_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:35_1_1_6_1_3_2_1_1_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E88">A403</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_6_1_3_2_1_1_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E89">A404</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E3:35_1_1_6_1_3_2_1_1_1_1_3_1_1_2"/>
( <font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i1</font> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i1</font>,<font color="Maroon" title="c12">j1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E32:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E80">A369</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E88">A403</font></span></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_6_1_3_2_1_1_1_1_3_1_1_2"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E14:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E63">A290</font></span></a>, <a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E81">A370</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E84">A397</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E86">A399</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E88">A403</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E89">A404</font></span></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font></span><br/><a NAME="E5:35_1_1_6_1_3_2_1_1_1_1_3_1_1_2"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c13">i2</font>,<font color="Maroon" title="c14">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E88">A403</font></span></a>, <a class="ref" href="gobrd13.html#T37" title="GOBRD13:th.37">GOBRD13:37</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_6_1_3_2_1_1_1_1_3_1_1_2"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E14:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E63">A290</font></span></a>, <a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E81">A370</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E84">A397</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E86">A399</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E88">A403</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E89">A404</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_1_1_1_3_1_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:35_1_1_6_1_3_2_1_1_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E88">A405</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_6_1_3_2_1_1_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E89">A406</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E3:35_1_1_6_1_3_2_1_1_1_1_3_1_1_3"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E32:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E80">A369</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E88">A405</font></span></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_6_1_3_2_1_1_1_1_3_1_1_3"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E13:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E62">A289</font></span></a>, <a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E81">A370</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E82">A371</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E84">A397</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E86">A399</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E88">A405</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E89">A406</font></span></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font></span><br/><a NAME="E5:35_1_1_6_1_3_2_1_1_1_1_3_1_1_3"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E88">A405</font></span></a>, <a class="ref" href="gobrd13.html#T39" title="GOBRD13:th.39">GOBRD13:39</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_6_1_3_2_1_1_1_1_3_1_1_3"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E13:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E62">A289</font></span></a>, <a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E81">A370</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E82">A371</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E84">A397</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E86">A399</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E88">A405</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E89">A406</font></span></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_1_1_1_3_1_1_4"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:35_1_1_6_1_3_2_1_1_1_1_3_1_1_4"/><span class="lab"><font color="Green" title="E88">A407</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_6_1_3_2_1_1_1_1_3_1_1_4"/><span class="lab"><font color="Green" title="E89">A408</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E3:35_1_1_6_1_3_2_1_1_1_1_3_1_1_4"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c13">i2</font>,<font color="Maroon" title="c14">j2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E32:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E80">A369</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E88">A407</font></span></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_6_1_3_2_1_1_1_1_3_1_1_4"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E25:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E74">A301</font></span></a>, <a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E81">A370</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E84">A397</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E86">A399</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E88">A407</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E89">A408</font></span></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font></span><br/><a NAME="E5:35_1_1_6_1_3_2_1_1_1_1_3_1_1_4"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c13">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E56">A283</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E54">A281</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E57">A284</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E58">A285</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E60">A287</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E88">A407</font></span></a>, <a class="ref" href="gobrd13.html#T41" title="GOBRD13:th.41">GOBRD13:41</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_6_1_3_2_1_1_1_1_3_1_1_4"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="txt" href="#E9:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E59">A286</font></span></a>, <a class="txt" href="#E25:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E74">A301</font></span></a>, <a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1"><span class="lab"><font color="Green" title="E81">A370</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E84">A397</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E86">A399</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E88">A407</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1_3_2_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E89">A408</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:35_1_1_6_1_3_2_1_1_1_1_3"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_6_1_3_2_1_2"/>
<font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E2:35_1_1_6_1_3_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E79">A368</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:35_1_1_6_1_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A409</font></span>: 
<font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E6:35_1_1_6"><span class="lab"><font color="Green" title="E51">A238</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">i1</font>, <font color="Maroon" title="c17">j1</font>, <font color="Maroon" title="c18">i2</font>, <font color="Maroon" title="c19">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:35_1_1_6_1_3_2_1_2"/><span class="lab"><font color="Green" title="E82">A410</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i1</font>,<font color="Maroon" title="c17">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:35_1_1_6_1_3_2_1_2"/><span class="lab"><font color="Green" title="E83">A411</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i1</font>,<font color="Maroon" title="c17">j1</font>
 <span class="kw">and </span><br/><a NAME="E7:35_1_1_6_1_3_2_1_2"/><span class="lab"><font color="Green" title="E84">A412</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i2</font>,<font color="Maroon" title="c19">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:35_1_1_6_1_3_2_1_2"/><span class="lab"><font color="Green" title="E85">A413</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i2</font>,<font color="Maroon" title="c19">j2</font>
 <span class="kw">and </span><br/><a NAME="E9:35_1_1_6_1_3_2_1_2"/><span class="lab"><font color="Green" title="E86">A414</font></span>: 
( ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>;<br/><a NAME="E10:35_1_1_6_1_3_2_1_2"/><span class="lab"><font color="Green" title="E87">A415</font></span>: 
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_6"><span class="lab"><font color="Green" title="E49">A236</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E81">A409</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E88">A416</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_6_1_3_2_1_2_1_1"/>
( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> ) or (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_6_1_3_2_1_2_1_1_1"/>
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">i</font>, <font color="Maroon" title="c21">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:35_1_1_6_1_3_2_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_6_1_3_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E89">A417</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a>, <a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E20:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E69">A296</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c22">i</font> = <font color="Maroon" title="c20">i</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c22">i</font>,<font color="Olive" title="b1">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">take </span><font color="Maroon" title="c23">j</font> = <font color="Maroon" title="c21">j</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c23">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">thus </span><a NAME="E5:35_1_1_6_1_3_2_1_2_1_1_1"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c23">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_6_1_3_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E89">A417</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_6_1_3_2_1_2_1_1_2"/>
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">i</font>, <font color="Maroon" title="c21">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:35_1_1_6_1_3_2_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_6_1_3_2_1_2_1_1_2"/><span class="lab"><font color="Green" title="E89">A418</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a>, <a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E20:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E69">A296</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c22">i</font> = <font color="Maroon" title="c20">i</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c22">i</font>,<font color="Olive" title="b1">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">take </span><font color="Maroon" title="c23">j</font> = <font color="Maroon" title="c21">j</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c23">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">thus </span><a NAME="E5:35_1_1_6_1_3_2_1_2_1_1_2"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c23">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_6_1_3_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E89">A418</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_6_1_3_2_1_2_1_1_3"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">i</font>, <font color="Maroon" title="c21">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:35_1_1_6_1_3_2_1_2_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_6_1_3_2_1_2_1_1_3"/><span class="lab"><font color="Green" title="E89">A419</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a>, <a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E5:35_1_1_6"><span class="lab"><font color="Green" title="E50">A237</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E52">A279</font></span></a>, <a class="txt" href="#E20:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E69">A296</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c22">i</font> = <font color="Maroon" title="c20">i</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c22">i</font>,<font color="Olive" title="b1">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">take </span><font color="Maroon" title="c23">j</font> = <font color="Maroon" title="c21">j</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c23">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">thus </span><a NAME="E5:35_1_1_6_1_3_2_1_2_1_1_3"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c23">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_6_1_3_2_1_2_1_1_3"><span class="lab"><font color="Green" title="E89">A419</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E12:35_1_1_6_1_3_2_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E13:35_1_1_6_1_3_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E81">A409</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E14:35_1_1_6_1_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A420</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i2</font>,<font color="Maroon" title="c19">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E85">A413</font></span></a>, <a class="txt" href="#E11:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E88">A416</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/><a NAME="E15:35_1_1_6_1_3_2_1_2"/>
<font color="Maroon" title="c15">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E81">A409</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E16:35_1_1_6_1_3_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E17:35_1_1_6_1_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A421</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i1</font>,<font color="Maroon" title="c17">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E83">A411</font></span></a>, <a class="txt" href="#E11:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E88">A416</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_2_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_2_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_6_1_3_2_1_2_2_1"/>
( ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E86">A414</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_6_1_3_2_1_2_2_1_1"/><span class="lab"><font color="Green" title="E91">A422</font></span>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E2:35_1_1_6_1_3_2_1_2_2_1_1"/><span class="kw">then </span>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j1</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c16">i1</font>,<font color="Maroon" title="c17">j1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E81">A409</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E82">A410</font></span></a>, <a class="txt" href="#E6:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E83">A411</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E84">A412</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E85">A413</font></span></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_6_1_3_2_1_2_2_1_1"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E31:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E79">A368</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E82">A410</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E84">A412</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E87">A415</font></span></a>, <a class="txt" href="#E17:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E90">A421</font></span></a>, <a class="txt" href="#E14:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E89">A420</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_2_2_1_1"><span class="lab"><font color="Green" title="E91">A422</font></span></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_6_1_3_2_1_2_2_1_2"/><span class="lab"><font color="Green" title="E91">A423</font></span>: 
( <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E2:35_1_1_6_1_3_2_1_2_2_1_2"/><span class="kw">then </span>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c16">i1</font>,<font color="Maroon" title="c17">j1</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c16">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E81">A409</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E82">A410</font></span></a>, <a class="txt" href="#E6:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E83">A411</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E84">A412</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E85">A413</font></span></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_6_1_3_2_1_2_2_1_2"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E31:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E79">A368</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E82">A410</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E84">A412</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E87">A415</font></span></a>, <a class="txt" href="#E17:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E90">A421</font></span></a>, <a class="txt" href="#E14:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E89">A420</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_2_2_1_2"><span class="lab"><font color="Green" title="E91">A423</font></span></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_2_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_6_1_3_2_1_2_2_1_3"/><span class="lab"><font color="Green" title="E91">A424</font></span>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E2:35_1_1_6_1_3_2_1_2_2_1_3"/><span class="kw">then </span>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c18">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c18">i2</font>,<font color="Maroon" title="c19">j2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E81">A409</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E82">A410</font></span></a>, <a class="txt" href="#E6:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E83">A411</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E84">A412</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E85">A413</font></span></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_6_1_3_2_1_2_2_1_3"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E31:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E79">A368</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E82">A410</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E84">A412</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E87">A415</font></span></a>, <a class="txt" href="#E17:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E90">A421</font></span></a>, <a class="txt" href="#E14:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E89">A420</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_2_2_1_3"><span class="lab"><font color="Green" title="E91">A424</font></span></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_6_1_3_2_1_2_2_1_4"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_6_1_3_2_1_2_2_1_4"/><span class="lab"><font color="Green" title="E91">A425</font></span>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E2:35_1_1_6_1_3_2_1_2_2_1_4"/><span class="kw">then </span>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c18">i2</font>,<font color="Maroon" title="c19">j2</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_6"><span class="lab"><font color="Green" title="E48">A235</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E3:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E81">A409</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E82">A410</font></span></a>, <a class="txt" href="#E6:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E83">A411</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E84">A412</font></span></a>, <a class="txt" href="#E8:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E85">A413</font></span></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_6_1_3_2_1_2_2_1_4"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E77">A304</font></span></a>, <a class="txt" href="#E30:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E78">A367</font></span></a>, <a class="txt" href="#E31:35_1_1_6_1_3"><span class="lab"><font color="Green" title="E79">A368</font></span></a>, <a class="txt" href="#E5:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E82">A410</font></span></a>, <a class="txt" href="#E7:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E84">A412</font></span></a>, <a class="txt" href="#E10:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E87">A415</font></span></a>, <a class="txt" href="#E17:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E90">A421</font></span></a>, <a class="txt" href="#E14:35_1_1_6_1_3_2_1_2"><span class="lab"><font color="Green" title="E89">A420</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1_3_2_1_2_2_1_4"><span class="lab"><font color="Green" title="E91">A425</font></span></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E19:35_1_1_6_1_3_2_1_2"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E34:35_1_1_6_1_3"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c15">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">defpred </span><font color="Maroon">S<sub>5</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> ex <font color="Olive" title="b1">w</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; $1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  ex <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">w</font></span>)</span> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">w</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">w</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">w</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">w</font></span>)</span></span>)</span> ) );<br/>
<a NAME="E19:35_1_1"/><span class="lab"><font color="Green" title="E46">A426</font></span>: 
<font color="Maroon">S<sub>4</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_7"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_1_1_7"/>
( ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font> ) ) &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> holds <br/>for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">i</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b5">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1"><span class="lab"><font color="Green" title="E34">A170</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:35_1_1_7"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> holds <br/>(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c10">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font> &amp; <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> implies (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c10">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c10">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:35_1_1_7"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_7"/><span class="lab"><font color="Green" title="E47">A427</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c10">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c10">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/>

<span class="kw">thus </span><a NAME="E5:35_1_1_7"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c10">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c10">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1"><span class="lab"><font color="Green" title="E34">A170</font></span></a>, <a class="txt" href="#E4:35_1_1_7"><span class="lab"><font color="Green" title="E47">A427</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E20:35_1_1"/><span class="lab"><font color="Green" title="E47">A428</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E19:35_1_1"><span class="lab"><font color="Green" title="E46">A426</font></span></a>, <a class="txt" href="#E18:35_1_1"><span class="lab"><font color="Green" title="E45">A232</font></span></a>);<br/></span>
<a NAME="E21:35_1_1"/><span class="lab"><font color="Green" title="E48">A429</font></span>: 
for <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j1</font>, <font color="Olive" title="b5">j2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i1</font>,<font color="Olive" title="b4">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i1</font>,<font color="Olive" title="b4">j1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i2</font>,<font color="Olive" title="b5">j2</font> holds <br/>( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies ( <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i2</font>,<span class="p3">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i2</font>,<span class="p3">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) ) ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies ( <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i2</font>,<span class="p3">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i2</font>,<span class="p3">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) ) ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies ( <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i2</font>,<span class="p3">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i2</font>,<span class="p3">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">k</font>, <font color="Maroon" title="c11">i1</font>, <font color="Maroon" title="c12">i2</font>, <font color="Maroon" title="c13">j1</font>, <font color="Maroon" title="c14">j2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i1</font>,<font color="Maroon" title="c13">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i1</font>,<font color="Maroon" title="c13">j1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<font color="Maroon" title="c14">j2</font> implies ( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies ( <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) ) ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies ( <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) ) ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies ( <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) ) ) ) )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_1_8"/><span class="lab"><font color="Green" title="E49">A430</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_8"/><span class="lab"><font color="Green" title="E50">A431</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i1</font>,<font color="Maroon" title="c13">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:35_1_1_8"/><span class="lab"><font color="Green" title="E51">A432</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i1</font>,<font color="Maroon" title="c13">j1</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_8"/><span class="lab"><font color="Green" title="E52">A433</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:35_1_1_8"/><span class="lab"><font color="Green" title="E53">A434</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<font color="Maroon" title="c14">j2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies ( <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) ) ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies ( <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) ) ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies ( <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) ) ) )</span><br/>

<a NAME="E6:35_1_1_8"/><span class="lab"><font color="Green" title="E54">A435</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a></span>;<br/>
<a NAME="E7:35_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A436</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_8"><span class="lab"><font color="Green" title="E49">A430</font></span></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/>
<a NAME="E8:35_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A437</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E9:35_1_1_8"/><span class="lab"><font color="Green" title="E57">A438</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_8"><span class="lab"><font color="Green" title="E49">A430</font></span></a>, <a class="txt" href="#E6:35_1_1_8"><span class="lab"><font color="Green" title="E54">A435</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a></span>;<br/>
<a NAME="E10:35_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A439</font></span>: 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E7:35_1_1_8"><span class="lab"><font color="Green" title="E55">A436</font></span></a></span>;<br/>
<a NAME="E11:35_1_1_8"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T35" title="NAT_D:th.35">NAT_D:35</a></span>;<br/>
<a NAME="E12:35_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A440</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_8"><span class="lab"><font color="Green" title="E57">A438</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E13:35_1_1_8"/><span class="lab"><font color="Green" title="E60">A441</font></span>: 
( <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">j1</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">j2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E14:35_1_1_8"/><span class="lab"><font color="Green" title="E61">A442</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a></span>;<br/>
<a NAME="E15:35_1_1_8"/><span class="lab"><font color="Green" title="E62">A443</font></span>: 
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">i1</font> &amp; <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">i2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E16:35_1_1_8"/><span class="lab"><font color="Green" title="E63">A444</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_8"><span class="lab"><font color="Green" title="E49">A430</font></span></a>, <a class="txt" href="#E6:35_1_1_8"><span class="lab"><font color="Green" title="E54">A435</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies ( <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) ) ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies ( <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) ) ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_8_1"/>
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">i</font>, <font color="Maroon" title="c16">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:35_1_1_8_1"/><span class="lab"><font color="Green" title="E64">A445</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_8_1"/><span class="lab"><font color="Green" title="E65">A446</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a>, <a class="txt" href="#E1:35_1_1_8"><span class="lab"><font color="Green" title="E49">A430</font></span></a>, <a class="txt" href="#E14:35_1_1_8"><span class="lab"><font color="Green" title="E61">A442</font></span></a>, <a class="txt" href="#E6:35_1_1_8"><span class="lab"><font color="Green" title="E54">A435</font></span></a>, <a class="txt" href="#E10:35_1_1_8"><span class="lab"><font color="Green" title="E58">A439</font></span></a></span>;<br/><span class="kw">thus </span><a NAME="E5:35_1_1_8_1"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_8_1"><span class="lab"><font color="Green" title="E64">A445</font></span></a>, <a class="txt" href="#E4:35_1_1_8_1"><span class="lab"><font color="Green" title="E65">A446</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )</span><br/><a NAME="E6:35_1_1_8_1"/><span class="lab"><font color="Green" title="E66">A447</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_8_1"><span class="lab"><font color="Green" title="E65">A446</font></span></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></span>;<br/><a NAME="E7:35_1_1_8_1"/><span class="lab"><font color="Green" title="E67">A448</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i1</font>,<font color="Maroon" title="c13">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<font color="Maroon" title="c14">j2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_8"><span class="lab"><font color="Green" title="E51">A432</font></span></a>, <a class="txt" href="#E5:35_1_1_8"><span class="lab"><font color="Green" title="E53">A434</font></span></a>, <a class="txt" href="#E12:35_1_1_8"><span class="lab"><font color="Green" title="E59">A440</font></span></a>, <a class="txt" href="#E16:35_1_1_8"><span class="lab"><font color="Green" title="E63">A444</font></span></a>, <a class="txt" href="#E4:35_1_1_8_1"><span class="lab"><font color="Green" title="E65">A446</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/><span class="kw">hence </span><a NAME="E8:35_1_1_8_1"/>
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_8"><span class="lab"><font color="Green" title="E50">A431</font></span></a>, <a class="txt" href="#E4:35_1_1_8"><span class="lab"><font color="Green" title="E52">A433</font></span></a>, <a class="txt" href="#E7:35_1_1_8"><span class="lab"><font color="Green" title="E55">A436</font></span></a>, <a class="txt" href="#E8:35_1_1_8"><span class="lab"><font color="Green" title="E56">A437</font></span></a>, <a class="txt" href="#E13:35_1_1_8"><span class="lab"><font color="Green" title="E60">A441</font></span></a>, <a class="txt" href="#E3:35_1_1_8_1"><span class="lab"><font color="Green" title="E64">A445</font></span></a>, <a class="txt" href="#E4:35_1_1_8_1"><span class="lab"><font color="Green" title="E65">A446</font></span></a>, <a class="txt" href="#E6:35_1_1_8_1"><span class="lab"><font color="Green" title="E66">A447</font></span></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )</span><br/><span class="kw">thus </span><a NAME="E9:35_1_1_8_1"/>
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_8"><span class="lab"><font color="Green" title="E50">A431</font></span></a>, <a class="txt" href="#E4:35_1_1_8"><span class="lab"><font color="Green" title="E52">A433</font></span></a>, <a class="txt" href="#E7:35_1_1_8"><span class="lab"><font color="Green" title="E55">A436</font></span></a>, <a class="txt" href="#E8:35_1_1_8"><span class="lab"><font color="Green" title="E56">A437</font></span></a>, <a class="txt" href="#E15:35_1_1_8"><span class="lab"><font color="Green" title="E62">A443</font></span></a>, <a class="txt" href="#E3:35_1_1_8_1"><span class="lab"><font color="Green" title="E64">A445</font></span></a>, <a class="txt" href="#E4:35_1_1_8_1"><span class="lab"><font color="Green" title="E65">A446</font></span></a>, <a class="txt" href="#E7:35_1_1_8_1"><span class="lab"><font color="Green" title="E67">A448</font></span></a>, <a class="txt" href="#E6:35_1_1_8_1"><span class="lab"><font color="Green" title="E66">A447</font></span></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )</span><br/><span class="kw">thus </span><a NAME="E10:35_1_1_8_1"/>
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_8"><span class="lab"><font color="Green" title="E50">A431</font></span></a>, <a class="txt" href="#E4:35_1_1_8"><span class="lab"><font color="Green" title="E52">A433</font></span></a>, <a class="txt" href="#E7:35_1_1_8"><span class="lab"><font color="Green" title="E55">A436</font></span></a>, <a class="txt" href="#E8:35_1_1_8"><span class="lab"><font color="Green" title="E56">A437</font></span></a>, <a class="txt" href="#E15:35_1_1_8"><span class="lab"><font color="Green" title="E62">A443</font></span></a>, <a class="txt" href="#E3:35_1_1_8_1"><span class="lab"><font color="Green" title="E64">A445</font></span></a>, <a class="txt" href="#E4:35_1_1_8_1"><span class="lab"><font color="Green" title="E65">A446</font></span></a>, <a class="txt" href="#E7:35_1_1_8_1"><span class="lab"><font color="Green" title="E67">A448</font></span></a>, <a class="txt" href="#E6:35_1_1_8_1"><span class="lab"><font color="Green" title="E66">A447</font></span></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )</span><br/><span class="kw">thus </span><a NAME="E11:35_1_1_8_1"/>
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_8"><span class="lab"><font color="Green" title="E50">A431</font></span></a>, <a class="txt" href="#E4:35_1_1_8"><span class="lab"><font color="Green" title="E52">A433</font></span></a>, <a class="txt" href="#E7:35_1_1_8"><span class="lab"><font color="Green" title="E55">A436</font></span></a>, <a class="txt" href="#E8:35_1_1_8"><span class="lab"><font color="Green" title="E56">A437</font></span></a>, <a class="txt" href="#E13:35_1_1_8"><span class="lab"><font color="Green" title="E60">A441</font></span></a>, <a class="txt" href="#E3:35_1_1_8_1"><span class="lab"><font color="Green" title="E64">A445</font></span></a>, <a class="txt" href="#E4:35_1_1_8_1"><span class="lab"><font color="Green" title="E65">A446</font></span></a>, <a class="txt" href="#E7:35_1_1_8_1"><span class="lab"><font color="Green" title="E67">A448</font></span></a>, <a class="txt" href="#E6:35_1_1_8_1"><span class="lab"><font color="Green" title="E66">A447</font></span></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies ( <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_8_2"/>
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">i</font>, <font color="Maroon" title="c16">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:35_1_1_8_2"/><span class="lab"><font color="Green" title="E64">A449</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_8_2"/><span class="lab"><font color="Green" title="E65">A450</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a>, <a class="txt" href="#E1:35_1_1_8"><span class="lab"><font color="Green" title="E49">A430</font></span></a>, <a class="txt" href="#E14:35_1_1_8"><span class="lab"><font color="Green" title="E61">A442</font></span></a>, <a class="txt" href="#E6:35_1_1_8"><span class="lab"><font color="Green" title="E54">A435</font></span></a>, <a class="txt" href="#E10:35_1_1_8"><span class="lab"><font color="Green" title="E58">A439</font></span></a></span>;<br/><span class="kw">thus </span><a NAME="E5:35_1_1_8_2"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_8_2"><span class="lab"><font color="Green" title="E64">A449</font></span></a>, <a class="txt" href="#E4:35_1_1_8_2"><span class="lab"><font color="Green" title="E65">A450</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )</span><br/><a NAME="E6:35_1_1_8_2"/><span class="lab"><font color="Green" title="E66">A451</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_8_2"><span class="lab"><font color="Green" title="E65">A450</font></span></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></span>;<br/><a NAME="E7:35_1_1_8_2"/><span class="lab"><font color="Green" title="E67">A452</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i1</font>,<font color="Maroon" title="c13">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<font color="Maroon" title="c14">j2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_8"><span class="lab"><font color="Green" title="E51">A432</font></span></a>, <a class="txt" href="#E5:35_1_1_8"><span class="lab"><font color="Green" title="E53">A434</font></span></a>, <a class="txt" href="#E12:35_1_1_8"><span class="lab"><font color="Green" title="E59">A440</font></span></a>, <a class="txt" href="#E16:35_1_1_8"><span class="lab"><font color="Green" title="E63">A444</font></span></a>, <a class="txt" href="#E4:35_1_1_8_2"><span class="lab"><font color="Green" title="E65">A450</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/><span class="kw">hence </span><a NAME="E8:35_1_1_8_2"/>
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_8"><span class="lab"><font color="Green" title="E50">A431</font></span></a>, <a class="txt" href="#E4:35_1_1_8"><span class="lab"><font color="Green" title="E52">A433</font></span></a>, <a class="txt" href="#E7:35_1_1_8"><span class="lab"><font color="Green" title="E55">A436</font></span></a>, <a class="txt" href="#E8:35_1_1_8"><span class="lab"><font color="Green" title="E56">A437</font></span></a>, <a class="txt" href="#E13:35_1_1_8"><span class="lab"><font color="Green" title="E60">A441</font></span></a>, <a class="txt" href="#E3:35_1_1_8_2"><span class="lab"><font color="Green" title="E64">A449</font></span></a>, <a class="txt" href="#E4:35_1_1_8_2"><span class="lab"><font color="Green" title="E65">A450</font></span></a>, <a class="txt" href="#E6:35_1_1_8_2"><span class="lab"><font color="Green" title="E66">A451</font></span></a>, <a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )</span><br/><span class="kw">thus </span><a NAME="E9:35_1_1_8_2"/>
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_8"><span class="lab"><font color="Green" title="E50">A431</font></span></a>, <a class="txt" href="#E4:35_1_1_8"><span class="lab"><font color="Green" title="E52">A433</font></span></a>, <a class="txt" href="#E7:35_1_1_8"><span class="lab"><font color="Green" title="E55">A436</font></span></a>, <a class="txt" href="#E8:35_1_1_8"><span class="lab"><font color="Green" title="E56">A437</font></span></a>, <a class="txt" href="#E15:35_1_1_8"><span class="lab"><font color="Green" title="E62">A443</font></span></a>, <a class="txt" href="#E3:35_1_1_8_2"><span class="lab"><font color="Green" title="E64">A449</font></span></a>, <a class="txt" href="#E4:35_1_1_8_2"><span class="lab"><font color="Green" title="E65">A450</font></span></a>, <a class="txt" href="#E7:35_1_1_8_2"><span class="lab"><font color="Green" title="E67">A452</font></span></a>, <a class="txt" href="#E6:35_1_1_8_2"><span class="lab"><font color="Green" title="E66">A451</font></span></a>, <a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )</span><br/><span class="kw">thus </span><a NAME="E10:35_1_1_8_2"/>
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_8"><span class="lab"><font color="Green" title="E50">A431</font></span></a>, <a class="txt" href="#E4:35_1_1_8"><span class="lab"><font color="Green" title="E52">A433</font></span></a>, <a class="txt" href="#E7:35_1_1_8"><span class="lab"><font color="Green" title="E55">A436</font></span></a>, <a class="txt" href="#E8:35_1_1_8"><span class="lab"><font color="Green" title="E56">A437</font></span></a>, <a class="txt" href="#E15:35_1_1_8"><span class="lab"><font color="Green" title="E62">A443</font></span></a>, <a class="txt" href="#E3:35_1_1_8_2"><span class="lab"><font color="Green" title="E64">A449</font></span></a>, <a class="txt" href="#E4:35_1_1_8_2"><span class="lab"><font color="Green" title="E65">A450</font></span></a>, <a class="txt" href="#E7:35_1_1_8_2"><span class="lab"><font color="Green" title="E67">A452</font></span></a>, <a class="txt" href="#E6:35_1_1_8_2"><span class="lab"><font color="Green" title="E66">A451</font></span></a>, <a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )</span><br/><span class="kw">thus </span><a NAME="E11:35_1_1_8_2"/>
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_8"><span class="lab"><font color="Green" title="E50">A431</font></span></a>, <a class="txt" href="#E4:35_1_1_8"><span class="lab"><font color="Green" title="E52">A433</font></span></a>, <a class="txt" href="#E7:35_1_1_8"><span class="lab"><font color="Green" title="E55">A436</font></span></a>, <a class="txt" href="#E8:35_1_1_8"><span class="lab"><font color="Green" title="E56">A437</font></span></a>, <a class="txt" href="#E13:35_1_1_8"><span class="lab"><font color="Green" title="E60">A441</font></span></a>, <a class="txt" href="#E3:35_1_1_8_2"><span class="lab"><font color="Green" title="E64">A449</font></span></a>, <a class="txt" href="#E4:35_1_1_8_2"><span class="lab"><font color="Green" title="E65">A450</font></span></a>, <a class="txt" href="#E7:35_1_1_8_2"><span class="lab"><font color="Green" title="E67">A452</font></span></a>, <a class="txt" href="#E6:35_1_1_8_2"><span class="lab"><font color="Green" title="E66">A451</font></span></a>, <a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E19:35_1_1_8"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">i</font>, <font color="Maroon" title="c16">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E21:35_1_1_8"/><span class="lab"><font color="Green" title="E64">A453</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">and </span><br/><a NAME="E22:35_1_1_8"/><span class="lab"><font color="Green" title="E65">A454</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a>, <a class="txt" href="#E1:35_1_1_8"><span class="lab"><font color="Green" title="E49">A430</font></span></a>, <a class="txt" href="#E14:35_1_1_8"><span class="lab"><font color="Green" title="E61">A442</font></span></a>, <a class="txt" href="#E6:35_1_1_8"><span class="lab"><font color="Green" title="E54">A435</font></span></a>, <a class="txt" href="#E10:35_1_1_8"><span class="lab"><font color="Green" title="E58">A439</font></span></a></span>;<br/>
<a NAME="E23:35_1_1_8"/><span class="lab"><font color="Green" title="E66">A455</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<font color="Maroon" title="c14">j2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_8"><span class="lab"><font color="Green" title="E53">A434</font></span></a>, <a class="txt" href="#E16:35_1_1_8"><span class="lab"><font color="Green" title="E63">A444</font></span></a>, <a class="txt" href="#E22:35_1_1_8"><span class="lab"><font color="Green" title="E65">A454</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<span class="kw">thus </span><a NAME="E24:35_1_1_8"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1_8"><span class="lab"><font color="Green" title="E64">A453</font></span></a>, <a class="txt" href="#E22:35_1_1_8"><span class="lab"><font color="Green" title="E65">A454</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )</span><br/>

<a NAME="E25:35_1_1_8"/><span class="lab"><font color="Green" title="E67">A456</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c16">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_1_1_8"><span class="lab"><font color="Green" title="E65">A454</font></span></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></span>;<br/>
<a NAME="E26:35_1_1_8"/><span class="lab"><font color="Green" title="E68">A457</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i1</font>,<font color="Maroon" title="c13">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_8"><span class="lab"><font color="Green" title="E51">A432</font></span></a>, <a class="txt" href="#E12:35_1_1_8"><span class="lab"><font color="Green" title="E59">A440</font></span></a>, <a class="txt" href="#E22:35_1_1_8"><span class="lab"><font color="Green" title="E65">A454</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:35_1_1_8"/>
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_8"><span class="lab"><font color="Green" title="E50">A431</font></span></a>, <a class="txt" href="#E4:35_1_1_8"><span class="lab"><font color="Green" title="E52">A433</font></span></a>, <a class="txt" href="#E7:35_1_1_8"><span class="lab"><font color="Green" title="E55">A436</font></span></a>, <a class="txt" href="#E8:35_1_1_8"><span class="lab"><font color="Green" title="E56">A437</font></span></a>, <a class="txt" href="#E13:35_1_1_8"><span class="lab"><font color="Green" title="E60">A441</font></span></a>, <a class="txt" href="#E21:35_1_1_8"><span class="lab"><font color="Green" title="E64">A453</font></span></a>, <a class="txt" href="#E22:35_1_1_8"><span class="lab"><font color="Green" title="E65">A454</font></span></a>, <a class="txt" href="#E23:35_1_1_8"><span class="lab"><font color="Green" title="E66">A455</font></span></a>, <a class="txt" href="#E25:35_1_1_8"><span class="lab"><font color="Green" title="E67">A456</font></span></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E28:35_1_1_8"/>
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_8"><span class="lab"><font color="Green" title="E50">A431</font></span></a>, <a class="txt" href="#E4:35_1_1_8"><span class="lab"><font color="Green" title="E52">A433</font></span></a>, <a class="txt" href="#E7:35_1_1_8"><span class="lab"><font color="Green" title="E55">A436</font></span></a>, <a class="txt" href="#E8:35_1_1_8"><span class="lab"><font color="Green" title="E56">A437</font></span></a>, <a class="txt" href="#E15:35_1_1_8"><span class="lab"><font color="Green" title="E62">A443</font></span></a>, <a class="txt" href="#E21:35_1_1_8"><span class="lab"><font color="Green" title="E64">A453</font></span></a>, <a class="txt" href="#E22:35_1_1_8"><span class="lab"><font color="Green" title="E65">A454</font></span></a>, <a class="txt" href="#E26:35_1_1_8"><span class="lab"><font color="Green" title="E68">A457</font></span></a>, <a class="txt" href="#E23:35_1_1_8"><span class="lab"><font color="Green" title="E66">A455</font></span></a>, <a class="txt" href="#E25:35_1_1_8"><span class="lab"><font color="Green" title="E67">A456</font></span></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E29:35_1_1_8"/>
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_8"><span class="lab"><font color="Green" title="E50">A431</font></span></a>, <a class="txt" href="#E4:35_1_1_8"><span class="lab"><font color="Green" title="E52">A433</font></span></a>, <a class="txt" href="#E7:35_1_1_8"><span class="lab"><font color="Green" title="E55">A436</font></span></a>, <a class="txt" href="#E8:35_1_1_8"><span class="lab"><font color="Green" title="E56">A437</font></span></a>, <a class="txt" href="#E15:35_1_1_8"><span class="lab"><font color="Green" title="E62">A443</font></span></a>, <a class="txt" href="#E21:35_1_1_8"><span class="lab"><font color="Green" title="E64">A453</font></span></a>, <a class="txt" href="#E22:35_1_1_8"><span class="lab"><font color="Green" title="E65">A454</font></span></a>, <a class="txt" href="#E26:35_1_1_8"><span class="lab"><font color="Green" title="E68">A457</font></span></a>, <a class="txt" href="#E23:35_1_1_8"><span class="lab"><font color="Green" title="E66">A455</font></span></a>, <a class="txt" href="#E25:35_1_1_8"><span class="lab"><font color="Green" title="E67">A456</font></span></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )</span><br/>

<span class="kw">thus </span><a NAME="E30:35_1_1_8"/>
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_8"><span class="lab"><font color="Green" title="E50">A431</font></span></a>, <a class="txt" href="#E4:35_1_1_8"><span class="lab"><font color="Green" title="E52">A433</font></span></a>, <a class="txt" href="#E7:35_1_1_8"><span class="lab"><font color="Green" title="E55">A436</font></span></a>, <a class="txt" href="#E8:35_1_1_8"><span class="lab"><font color="Green" title="E56">A437</font></span></a>, <a class="txt" href="#E13:35_1_1_8"><span class="lab"><font color="Green" title="E60">A441</font></span></a>, <a class="txt" href="#E21:35_1_1_8"><span class="lab"><font color="Green" title="E64">A453</font></span></a>, <a class="txt" href="#E22:35_1_1_8"><span class="lab"><font color="Green" title="E65">A454</font></span></a>, <a class="txt" href="#E26:35_1_1_8"><span class="lab"><font color="Green" title="E68">A457</font></span></a>, <a class="txt" href="#E23:35_1_1_8"><span class="lab"><font color="Green" title="E66">A455</font></span></a>, <a class="txt" href="#E25:35_1_1_8"><span class="lab"><font color="Green" title="E67">A456</font></span></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E22:35_1_1"/><span class="lab"><font color="Green" title="E49">A458</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 holds <br/>( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 implies ( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1_1_9"/><span class="lab"><font color="Green" title="E50">A459</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) )</span><br/>

<a NAME="E2:35_1_1_9"/><span class="lab"><font color="Green" title="E51">A460</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a></span>;<br/>
<a NAME="E3:35_1_1_9"/><span class="lab"><font color="Green" title="E52">A461</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a></span>;<br/>
<a NAME="E4:35_1_1_9"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_9"><span class="lab"><font color="Green" title="E50">A459</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/>
<a NAME="E5:35_1_1_9"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">j1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font> &amp; ( ( <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> ) or ( <font color="Olive" title="b1">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> ) or ( <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> ) or ( <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_9"><span class="lab"><font color="Green" title="E51">A460</font></span></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:35_1_1_9"/>
( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_9"><span class="lab"><font color="Green" title="E50">A459</font></span></a>, <a class="txt" href="#E3:35_1_1_9"><span class="lab"><font color="Green" title="E52">A461</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">defpred </span><font color="Maroon">S<sub>6</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> $1</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">m</font>;<br/>
<a NAME="E23:35_1_1"/><span class="lab"><font color="Green" title="E50">A462</font></span>: 
<font color="Maroon">S<sub>6</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">m</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1_1_10"/><span class="lab"><font color="Green" title="E51">A463</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">m</font></span><br/>

<a NAME="E2:35_1_1_10"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">m</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E3:35_1_1_10"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1"><span class="lab"><font color="Green" title="E34">A170</font></span></a>, <a class="txt" href="#E1:35_1_1_10"><span class="lab"><font color="Green" title="E51">A463</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E24:35_1_1"/><span class="lab"><font color="Green" title="E51">A464</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/>

<a NAME="E1:35_1_1_11"/><span class="lab"><font color="Green" title="E52">A465</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a></span>;<br/>
<a NAME="E2:35_1_1_11"/><span class="lab"><font color="Green" title="E53">A466</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a></span>;<br/>
<a NAME="E3:35_1_1_11"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan8.html#T13" title="JORDAN8:th.13">JORDAN8:13</a></span>;<br/>
<a NAME="E4:35_1_1_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A467</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan8.html#D1" title="JORDAN8:def.1">JORDAN8:def 1</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_11_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_11_1"/>
( <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 or <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_11_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_11_1_1"/><span class="lab"><font color="Green" title="E55">A468</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c11">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:35_1_1_11_1_1"/><span class="lab"><font color="Green" title="E56">A469</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_11_1_1"/><span class="lab"><font color="Green" title="E57">A470</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:35_1_1_11_1_1"/>
 <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:35_1_1_11_1_1"/>
 <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:35_1_1_11_1_1"/><span class="lab"><font color="Green" title="E58">A471</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c11">i</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i</font>,<font color="Olive" title="b1">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,<font color="Olive" title="b1">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><span class="kw">take </span><font color="Maroon" title="c12">j</font> =  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><a NAME="E8:35_1_1_11_1_1"/>
<font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_11_1_1"><span class="lab"><font color="Green" title="E57">A470</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E9:35_1_1_11_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_11"><span class="lab"><font color="Green" title="E54">A467</font></span></a>, <a class="txt" href="#E3:35_1_1_11_1_1"><span class="lab"><font color="Green" title="E56">A469</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><a NAME="E10:35_1_1_11_1_1"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_11_1_1"><span class="lab"><font color="Green" title="E55">A468</font></span></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><span class="kw">hence </span><a NAME="E11:35_1_1_11_1_1"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1"><span class="lab"><font color="Green" title="E34">A170</font></span></a>, <a class="txt" href="#E7:35_1_1_11_1_1"><span class="lab"><font color="Green" title="E58">A471</font></span></a>, <a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_11_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_11_1_2"/><span class="lab"><font color="Green" title="E55">A472</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c11">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:35_1_1_11_1_2"/><span class="lab"><font color="Green" title="E56">A473</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_11_1_2"/><span class="lab"><font color="Green" title="E57">A474</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:35_1_1_11_1_2"/><span class="lab"><font color="Green" title="E58">A475</font></span>: 
(  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )
 <span class="kw">and </span><br/><a NAME="E6:35_1_1_11_1_2"/><span class="lab"><font color="Green" title="E59">A476</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><span class="kw">take </span><font color="Maroon" title="c12">j</font> =  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><a NAME="E7:35_1_1_11_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_11_1_2"><span class="lab"><font color="Green" title="E56">A473</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E8:35_1_1_11_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_11"><span class="lab"><font color="Green" title="E54">A467</font></span></a>, <a class="txt" href="#E4:35_1_1_11_1_2"><span class="lab"><font color="Green" title="E57">A474</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">consider </span><font color="Maroon" title="c13">i9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E10:35_1_1_11_1_2"/><span class="lab"><font color="Green" title="E60">A477</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">i9</font> &amp; <font color="Maroon" title="c13">i9</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp;  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c13">i9</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i9</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )
 <span class="kw">and </span><br/><a NAME="E11:35_1_1_11_1_2"/><span class="lab"><font color="Green" title="E61">A478</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i9</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">i9</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a></span>;<br/><a NAME="E12:35_1_1_11_1_2"/>
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_11_1_2"><span class="lab"><font color="Green" title="E56">A473</font></span></a>, <a class="txt" href="#E4:35_1_1_11_1_2"><span class="lab"><font color="Green" title="E57">A474</font></span></a>, <a class="txt" href="#E5:35_1_1_11_1_2"><span class="lab"><font color="Green" title="E58">A475</font></span></a>, <a class="txt" href="#E10:35_1_1_11_1_2"><span class="lab"><font color="Green" title="E60">A477</font></span></a>, <a class="ref" href="jordan9.html#T31" target="_self" title="JORDAN9:th.31">Th31</a></span>;<br/><span class="kw">hence </span><a NAME="E13:35_1_1_11_1_2"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_11_1_2"><span class="lab"><font color="Green" title="E55">A472</font></span></a>, <a class="txt" href="#E6:35_1_1_11_1_2"><span class="lab"><font color="Green" title="E59">A476</font></span></a>, <a class="txt" href="#E11:35_1_1_11_1_2"><span class="lab"><font color="Green" title="E61">A478</font></span></a>, <a class="ref" href="finseq_1.html#D9" title="FINSEQ_1:def.9">FINSEQ_1:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_11_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_11_1_3"/><span class="lab"><font color="Green" title="E55">A479</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><a NAME="E2:35_1_1_11_1_3"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_11"><span class="lab"><font color="Green" title="E53">A466</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">i1</font>, <font color="Maroon" title="c12">j1</font>, <font color="Maroon" title="c13">i2</font>, <font color="Maroon" title="c14">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:35_1_1_11_1_3"/><span class="lab"><font color="Green" title="E56">A480</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i1</font>,<font color="Maroon" title="c12">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i1</font>,<font color="Maroon" title="c12">j1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<font color="Maroon" title="c14">j2</font> )
 <span class="kw">and </span><br/><a NAME="E5:35_1_1_11_1_3"/><span class="lab"><font color="Green" title="E57">A481</font></span>: 
( ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_11"><span class="lab"><font color="Green" title="E52">A465</font></span></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_11_1_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_11_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_11_1_3_1_1"/>
( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> ) or (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_11_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_11_1_3_1_1_1"/><span class="lab"><font color="Green" title="E58">A482</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_11_1_3_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_11_1_3_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_11_1_3_1_1_1_1_1"/>
( ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E57">A481</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_11_1_3_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_11_1_3_1_1_1_1_1_1"/>
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><a NAME="E2:35_1_1_11_1_3_1_1_1_1_1_1"/><span class="kw">then </span>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E55">A479</font></span></a>, <a class="txt" href="#E4:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E56">A480</font></span></a>, <a class="txt" href="#E1:35_1_1_11_1_3_1_1_1"><span class="lab"><font color="Green" title="E58">A482</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_11_1_3_1_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_11_1_3_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_11_1_3_1_1_1_1_1_2"/>
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><a NAME="E2:35_1_1_11_1_3_1_1_1_1_1_2"/><span class="kw">then </span>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E55">A479</font></span></a>, <a class="txt" href="#E4:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E56">A480</font></span></a>, <a class="txt" href="#E1:35_1_1_11_1_3_1_1_1"><span class="lab"><font color="Green" title="E58">A482</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_11_1_3_1_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_11_1_3_1_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_11_1_3_1_1_1_1_1_3"/>
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><a NAME="E2:35_1_1_11_1_3_1_1_1_1_1_3"/><span class="kw">then </span>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E55">A479</font></span></a>, <a class="txt" href="#E4:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E56">A480</font></span></a>, <a class="txt" href="#E1:35_1_1_11_1_3_1_1_1"><span class="lab"><font color="Green" title="E58">A482</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_11_1_3_1_1_1_1_1_3"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_11_1_3_1_1_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_11_1_3_1_1_1_1_1_4"/>
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><a NAME="E2:35_1_1_11_1_3_1_1_1_1_1_4"/><span class="kw">then </span>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E55">A479</font></span></a>, <a class="txt" href="#E4:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E56">A480</font></span></a>, <a class="txt" href="#E1:35_1_1_11_1_3_1_1_1"><span class="lab"><font color="Green" title="E58">A482</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_11_1_3_1_1_1_1_1_4"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_11_1_3_1_1_1"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_11_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_11_1_3_1_1_2"/><span class="lab"><font color="Green" title="E58">A483</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_11_1_3_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_11_1_3_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_11_1_3_1_1_2_1_1"/>
( ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E57">A481</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_11_1_3_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_11_1_3_1_1_2_1_1_1"/>
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><a NAME="E2:35_1_1_11_1_3_1_1_2_1_1_1"/><span class="kw">then </span>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E55">A479</font></span></a>, <a class="txt" href="#E4:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E56">A480</font></span></a>, <a class="txt" href="#E1:35_1_1_11_1_3_1_1_2"><span class="lab"><font color="Green" title="E58">A483</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_11_1_3_1_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_11_1_3_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_11_1_3_1_1_2_1_1_2"/>
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><a NAME="E2:35_1_1_11_1_3_1_1_2_1_1_2"/><span class="kw">then </span>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E55">A479</font></span></a>, <a class="txt" href="#E4:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E56">A480</font></span></a>, <a class="txt" href="#E1:35_1_1_11_1_3_1_1_2"><span class="lab"><font color="Green" title="E58">A483</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_11_1_3_1_1_2_1_1_2"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_11_1_3_1_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_11_1_3_1_1_2_1_1_3"/>
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><a NAME="E2:35_1_1_11_1_3_1_1_2_1_1_3"/><span class="kw">then </span>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E55">A479</font></span></a>, <a class="txt" href="#E4:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E56">A480</font></span></a>, <a class="txt" href="#E1:35_1_1_11_1_3_1_1_2"><span class="lab"><font color="Green" title="E58">A483</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_11_1_3_1_1_2_1_1_3"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_11_1_3_1_1_2_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_11_1_3_1_1_2_1_1_4"/>
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><a NAME="E2:35_1_1_11_1_3_1_1_2_1_1_4"/><span class="kw">then </span>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E55">A479</font></span></a>, <a class="txt" href="#E4:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E56">A480</font></span></a>, <a class="txt" href="#E1:35_1_1_11_1_3_1_1_2"><span class="lab"><font color="Green" title="E58">A483</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_11_1_3_1_1_2_1_1_4"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_11_1_3_1_1_2"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_11_1_3_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_11_1_3_1_1_3"/><span class="lab"><font color="Green" title="E58">A484</font></span>: 
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_11_1_3_1_1_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_11_1_3_1_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_11_1_3_1_1_3_1_1"/>
( ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E57">A481</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_11_1_3_1_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_11_1_3_1_1_3_1_1_1"/>
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><a NAME="E2:35_1_1_11_1_3_1_1_3_1_1_1"/><span class="kw">then </span>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E55">A479</font></span></a>, <a class="txt" href="#E4:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E56">A480</font></span></a>, <a class="txt" href="#E1:35_1_1_11_1_3_1_1_3"><span class="lab"><font color="Green" title="E58">A484</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_11_1_3_1_1_3_1_1_1"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_11_1_3_1_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_11_1_3_1_1_3_1_1_2"/>
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><a NAME="E2:35_1_1_11_1_3_1_1_3_1_1_2"/><span class="kw">then </span>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E55">A479</font></span></a>, <a class="txt" href="#E4:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E56">A480</font></span></a>, <a class="txt" href="#E1:35_1_1_11_1_3_1_1_3"><span class="lab"><font color="Green" title="E58">A484</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_11_1_3_1_1_3_1_1_2"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_11_1_3_1_1_3_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_11_1_3_1_1_3_1_1_3"/>
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><a NAME="E2:35_1_1_11_1_3_1_1_3_1_1_3"/><span class="kw">then </span>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E55">A479</font></span></a>, <a class="txt" href="#E4:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E56">A480</font></span></a>, <a class="txt" href="#E1:35_1_1_11_1_3_1_1_3"><span class="lab"><font color="Green" title="E58">A484</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_11_1_3_1_1_3_1_1_3"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_11_1_3_1_1_3_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_11_1_3_1_1_3_1_1_4"/>
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><a NAME="E2:35_1_1_11_1_3_1_1_3_1_1_4"/><span class="kw">then </span>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E55">A479</font></span></a>, <a class="txt" href="#E4:35_1_1_11_1_3"><span class="lab"><font color="Green" title="E56">A480</font></span></a>, <a class="txt" href="#E1:35_1_1_11_1_3_1_1_3"><span class="lab"><font color="Green" title="E58">A484</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_11_1_3_1_1_3_1_1_4"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_11_1_3_1_1_3"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:35_1_1_11_1_3"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E25:35_1_1"/><span class="lab"><font color="Green" title="E52">A485</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_12"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c10">k</font>] implies <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1_1_12"/><span class="lab"><font color="Green" title="E53">A486</font></span>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c11">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c11">m</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:35_1_1_12"/><span class="lab"><font color="Green" title="E54">A487</font></span>: 
<font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c11">m</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_12_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_12_1"/>
( <font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_12"><span class="lab"><font color="Green" title="E54">A487</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_12_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_12_1_1"/>
<font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c11">m</font></span><br/><div class="add"><a NAME="E2:35_1_1_12_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A488</font></span>: 
<font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:35_1_1_12_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> &amp;  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a>, <a class="txt" href="#E24:35_1_1"><span class="lab"><font color="Green" title="E51">A464</font></span></a></span>;<br/><a NAME="E4:35_1_1_12_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c11">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_12_1_1"><span class="lab"><font color="Green" title="E55">A488</font></span></a>, <a class="ref" href="finseq_5.html#T25" title="FINSEQ_5:th.25">FINSEQ_5:25</a></span>;<br/><span class="kw">hence </span><a NAME="E5:35_1_1_12_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c11">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_12"><span class="lab"><font color="Green" title="E53">A486</font></span></a>, <a class="txt" href="#E2:35_1_1_12_1_1"><span class="lab"><font color="Green" title="E55">A488</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_12_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_12_1_2"/><span class="lab"><font color="Green" title="E55">A489</font></span>: 
<font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c11">m</font></span><br/><div class="add"><a NAME="E2:35_1_1_12_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_12_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c11">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_12_1_2"><span class="lab"><font color="Green" title="E55">A489</font></span></a>, <a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E26:35_1_1"/><span class="lab"><font color="Green" title="E53">A490</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E23:35_1_1"><span class="lab"><font color="Green" title="E50">A462</font></span></a>, <a class="txt" href="#E25:35_1_1"><span class="lab"><font color="Green" title="E52">A485</font></span></a>);<br/></span>
<span class="kw">defpred </span><font color="Maroon">S<sub>7</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> $1 is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> ;<br/>
<a NAME="E27:35_1_1"/><span class="lab"><font color="Green" title="E54">A491</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>7</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_13"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font>] implies <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1_1_13"/><span class="lab"><font color="Green" title="E55">A492</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E2:35_1_1_13"/><span class="lab"><font color="Green" title="E56">A493</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_13_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_13_1"/>
( <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 or <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_13_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_13_1_1"/>
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:35_1_1_13_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E3:35_1_1_13_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_13_1_1"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sppol_2.html#T27" title="SPPOL_2:th.27">SPPOL_2:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_13_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_13_1_2"/><span class="lab"><font color="Green" title="E57">A494</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c11">m</font> = <font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1;<br/><a NAME="E2:35_1_1_13_1_2"/><span class="lab"><font color="Green" title="E58">A495</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E57">A494</font></span></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/><a NAME="E3:35_1_1_13_1_2"/><span class="lab"><font color="Green" title="E59">A496</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a></span>;<br/><a NAME="E4:35_1_1_13_1_2"/><span class="lab"><font color="Green" title="E60">A497</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E57">A494</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">i1</font>, <font color="Maroon" title="c13">j1</font>, <font color="Maroon" title="c14">i2</font>, <font color="Maroon" title="c15">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E6:35_1_1_13_1_2"/><span class="lab"><font color="Green" title="E61">A498</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i1</font>,<font color="Maroon" title="c13">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:35_1_1_13_1_2"/><span class="lab"><font color="Green" title="E62">A499</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i1</font>,<font color="Maroon" title="c13">j1</font>
 <span class="kw">and </span><br/><a NAME="E8:35_1_1_13_1_2"/><span class="lab"><font color="Green" title="E63">A500</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i2</font>,<font color="Maroon" title="c15">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:35_1_1_13_1_2"/><span class="lab"><font color="Green" title="E64">A501</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i2</font>,<font color="Maroon" title="c15">j2</font>
 <span class="kw">and </span><br/><a NAME="E10:35_1_1_13_1_2"/><span class="lab"><font color="Green" title="E65">A502</font></span>: 
( ( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> ) or ( <font color="Maroon" title="c12">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> ) or ( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> ) or ( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_13"><span class="lab"><font color="Green" title="E56">A493</font></span></a>, <a class="txt" href="#E2:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E58">A495</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>;<br/><a NAME="E11:35_1_1_13_1_2"/><span class="lab"><font color="Green" title="E66">A503</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i1</font>,<font color="Maroon" title="c13">j1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i2</font>,<font color="Maroon" title="c15">j2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E60">A497</font></span></a>, <a class="txt" href="#E2:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E58">A495</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E7:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E62">A499</font></span></a>, <a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></span>;<br/><a NAME="E12:35_1_1_13_1_2"/><span class="lab"><font color="Green" title="E67">A504</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E13:35_1_1_13_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A505</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/><a NAME="E14:35_1_1_13_1_2"/><span class="lab"><font color="Green" title="E69">A506</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E15:35_1_1_13_1_2"/><span class="lab"><font color="Green" title="E70">A507</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E16:35_1_1_13_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A508</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/><a NAME="E17:35_1_1_13_1_2"/><span class="lab"><font color="Green" title="E72">A509</font></span>: 
<font color="Maroon" title="c12">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E18:35_1_1_13_1_2"/><span class="lab"><font color="Green" title="E73">A510</font></span>: 
<font color="Maroon" title="c15">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E19:35_1_1_13_1_2"/><span class="lab"><font color="Green" title="E74">A511</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E20:35_1_1_13_1_2"/><span class="lab"><font color="Green" title="E75">A512</font></span>: 
<font color="Maroon" title="c13">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E21:35_1_1_13_1_2"/><span class="lab"><font color="Green" title="E76">A513</font></span>: 
<font color="Maroon" title="c14">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_13_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_13_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_13_1_2_1_1"/>
( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> ) or (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_13_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_13_1_2_1_1_1"/><span class="lab"><font color="Green" title="E77">A514</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_13_1_2_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_13_1_2_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_13_1_2_1_1_1_1_1"/>
( ( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> ) or ( <font color="Maroon" title="c12">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> ) or ( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> ) or ( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E65">A502</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_13_1_2_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_13_1_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E78">A515</font></span>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:35_1_1_13_1_2_1_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E57">A494</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="txt" href="#E7:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E62">A499</font></span></a>, <a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_1"><span class="lab"><font color="Green" title="E77">A514</font></span></a></span>;<br/><a NAME="E3:35_1_1_13_1_2_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E4:35_1_1_13_1_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A516</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E19:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E74">A511</font></span></a>, <a class="txt" href="#E14:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E69">A506</font></span></a>, <a class="txt" href="#E18:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E73">A510</font></span></a>, <a class="txt" href="#E11:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E66">A503</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E78">A515</font></span></a>, <a class="ref" href="goboard7.html#T17" title="GOBOARD7:th.17">GOBOARD7:17</a></span>;<br/><a NAME="E5:35_1_1_13_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E57">A494</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="txt" href="#E7:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E62">A499</font></span></a>, <a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_1"><span class="lab"><font color="Green" title="E77">A514</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E78">A515</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_13_1_2_1_1_1_1_1_1"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_13"><span class="lab"><font color="Green" title="E55">A492</font></span></a>, <a class="txt" href="#E2:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E58">A495</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E4:35_1_1_13_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E79">A516</font></span></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_13_1_2_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_13_1_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E78">A517</font></span>: 
( <font color="Maroon" title="c12">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:35_1_1_13_1_2_1_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E57">A494</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="txt" href="#E7:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E62">A499</font></span></a>, <a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_1"><span class="lab"><font color="Green" title="E77">A514</font></span></a></span>;<br/><a NAME="E3:35_1_1_13_1_2_1_1_1_1_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E4:35_1_1_13_1_2_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A518</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E19:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E74">A511</font></span></a>, <a class="txt" href="#E20:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E75">A512</font></span></a>, <a class="txt" href="#E21:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E76">A513</font></span></a>, <a class="txt" href="#E13:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E68">A505</font></span></a>, <a class="txt" href="#E11:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E66">A503</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E78">A517</font></span></a>, <a class="ref" href="goboard7.html#T18" title="GOBOARD7:th.18">GOBOARD7:18</a></span>;<br/><a NAME="E5:35_1_1_13_1_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E57">A494</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="txt" href="#E7:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E62">A499</font></span></a>, <a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_1"><span class="lab"><font color="Green" title="E77">A514</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E78">A517</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_13_1_2_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_13"><span class="lab"><font color="Green" title="E55">A492</font></span></a>, <a class="txt" href="#E2:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E58">A495</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E4:35_1_1_13_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E79">A518</font></span></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_13_1_2_1_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_13_1_2_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E78">A519</font></span>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:35_1_1_13_1_2_1_1_1_1_1_3"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E57">A494</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="txt" href="#E7:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E62">A499</font></span></a>, <a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_1"><span class="lab"><font color="Green" title="E77">A514</font></span></a></span>;<br/><a NAME="E3:35_1_1_13_1_2_1_1_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E4:35_1_1_13_1_2_1_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A520</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E17:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E72">A509</font></span></a>, <a class="txt" href="#E14:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E69">A506</font></span></a>, <a class="txt" href="#E15:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E70">A507</font></span></a>, <a class="txt" href="#E11:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E66">A503</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E78">A519</font></span></a>, <a class="ref" href="goboard7.html#T19" title="GOBOARD7:th.19">GOBOARD7:19</a></span>;<br/><a NAME="E5:35_1_1_13_1_2_1_1_1_1_1_3"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E57">A494</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="txt" href="#E7:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E62">A499</font></span></a>, <a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_1"><span class="lab"><font color="Green" title="E77">A514</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E78">A519</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_13_1_2_1_1_1_1_1_3"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_13"><span class="lab"><font color="Green" title="E55">A492</font></span></a>, <a class="txt" href="#E2:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E58">A495</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E4:35_1_1_13_1_2_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E79">A520</font></span></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_13_1_2_1_1_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_13_1_2_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E78">A521</font></span>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:35_1_1_13_1_2_1_1_1_1_1_4"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E57">A494</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="txt" href="#E7:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E62">A499</font></span></a>, <a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_1"><span class="lab"><font color="Green" title="E77">A514</font></span></a></span>;<br/><a NAME="E3:35_1_1_13_1_2_1_1_1_1_1_4"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E4:35_1_1_13_1_2_1_1_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A522</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E17:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E72">A509</font></span></a>, <a class="txt" href="#E20:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E75">A512</font></span></a>, <a class="txt" href="#E12:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E67">A504</font></span></a>, <a class="txt" href="#E16:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E71">A508</font></span></a>, <a class="txt" href="#E11:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E66">A503</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E78">A521</font></span></a>, <a class="ref" href="goboard7.html#T20" title="GOBOARD7:th.20">GOBOARD7:20</a></span>;<br/><a NAME="E5:35_1_1_13_1_2_1_1_1_1_1_4"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E57">A494</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="txt" href="#E7:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E62">A499</font></span></a>, <a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_1"><span class="lab"><font color="Green" title="E77">A514</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E78">A521</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_13_1_2_1_1_1_1_1_4"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_13"><span class="lab"><font color="Green" title="E55">A492</font></span></a>, <a class="txt" href="#E2:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E58">A495</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E4:35_1_1_13_1_2_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E79">A522</font></span></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_13_1_2_1_1_1"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_13_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_13_1_2_1_1_2"/><span class="lab"><font color="Green" title="E77">A523</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_13_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_13_1_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_13_1_2_1_1_2_1_1"/>
( ( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> ) or ( <font color="Maroon" title="c12">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> ) or ( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> ) or ( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E65">A502</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_13_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_13_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E78">A524</font></span>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:35_1_1_13_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E57">A494</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="txt" href="#E7:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E62">A499</font></span></a>, <a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_2"><span class="lab"><font color="Green" title="E77">A523</font></span></a></span>;<br/><a NAME="E3:35_1_1_13_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A525</font></span>: 
<font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E4:35_1_1_13_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_13_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E78">A524</font></span></a></span>;<br/><a NAME="E5:35_1_1_13_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A526</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E19:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E74">A511</font></span></a>, <a class="txt" href="#E17:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E72">A509</font></span></a>, <a class="txt" href="#E14:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E69">A506</font></span></a>, <a class="txt" href="#E11:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E66">A503</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E78">A524</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E79">A525</font></span></a>, <a class="ref" href="goboard7.html#T15" title="GOBOARD7:th.15">GOBOARD7:15</a></span>;<br/><a NAME="E6:35_1_1_13_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E57">A494</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="txt" href="#E7:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E62">A499</font></span></a>, <a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_2"><span class="lab"><font color="Green" title="E77">A523</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E78">A524</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:35_1_1_13_1_2_1_1_2_1_1_1"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_13"><span class="lab"><font color="Green" title="E55">A492</font></span></a>, <a class="txt" href="#E2:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E58">A495</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E5:35_1_1_13_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E80">A526</font></span></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_13_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_13_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E78">A527</font></span>: 
( <font color="Maroon" title="c12">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:35_1_1_13_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E57">A494</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="txt" href="#E7:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E62">A499</font></span></a>, <a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_2"><span class="lab"><font color="Green" title="E77">A523</font></span></a></span>;<br/><a NAME="E3:35_1_1_13_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A528</font></span>: 
<font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E4:35_1_1_13_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_13_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A527</font></span></a></span>;<br/><a NAME="E5:35_1_1_13_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A529</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E19:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E74">A511</font></span></a>, <a class="txt" href="#E14:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E69">A506</font></span></a>, <a class="txt" href="#E20:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E75">A512</font></span></a>, <a class="txt" href="#E11:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E66">A503</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A527</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E79">A528</font></span></a>, <a class="ref" href="goboard7.html#T16" title="GOBOARD7:th.16">GOBOARD7:16</a></span>;<br/><a NAME="E6:35_1_1_13_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E57">A494</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="txt" href="#E7:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E62">A499</font></span></a>, <a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_2"><span class="lab"><font color="Green" title="E77">A523</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A527</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:35_1_1_13_1_2_1_1_2_1_1_2"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_13"><span class="lab"><font color="Green" title="E55">A492</font></span></a>, <a class="txt" href="#E2:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E58">A495</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E5:35_1_1_13_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E80">A529</font></span></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_13_1_2_1_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_13_1_2_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E78">A530</font></span>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:35_1_1_13_1_2_1_1_2_1_1_3"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E57">A494</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="txt" href="#E7:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E62">A499</font></span></a>, <a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_2"><span class="lab"><font color="Green" title="E77">A523</font></span></a></span>;<br/><a NAME="E3:35_1_1_13_1_2_1_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A531</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E4:35_1_1_13_1_2_1_1_2_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><a NAME="E5:35_1_1_13_1_2_1_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A532</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E17:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E72">A509</font></span></a>, <a class="txt" href="#E14:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E69">A506</font></span></a>, <a class="txt" href="#E20:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E75">A512</font></span></a>, <a class="txt" href="#E16:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E71">A508</font></span></a>, <a class="txt" href="#E11:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E66">A503</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E78">A530</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E79">A531</font></span></a>, <a class="ref" href="goboard7.html#T16" title="GOBOARD7:th.16">GOBOARD7:16</a></span>;<br/><a NAME="E6:35_1_1_13_1_2_1_1_2_1_1_3"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E57">A494</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="txt" href="#E7:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E62">A499</font></span></a>, <a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_2"><span class="lab"><font color="Green" title="E77">A523</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E78">A530</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:35_1_1_13_1_2_1_1_2_1_1_3"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_13"><span class="lab"><font color="Green" title="E55">A492</font></span></a>, <a class="txt" href="#E2:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E58">A495</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E5:35_1_1_13_1_2_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E80">A532</font></span></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_13_1_2_1_1_2_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_13_1_2_1_1_2_1_1_4"/><span class="lab"><font color="Green" title="E78">A533</font></span>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:35_1_1_13_1_2_1_1_2_1_1_4"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E57">A494</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="txt" href="#E7:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E62">A499</font></span></a>, <a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_2"><span class="lab"><font color="Green" title="E77">A523</font></span></a></span>;<br/><a NAME="E3:35_1_1_13_1_2_1_1_2_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A534</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E4:35_1_1_13_1_2_1_1_2_1_1_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><a NAME="E5:35_1_1_13_1_2_1_1_2_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A535</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E19:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E74">A511</font></span></a>, <a class="txt" href="#E17:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E72">A509</font></span></a>, <a class="txt" href="#E20:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E75">A512</font></span></a>, <a class="txt" href="#E13:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E68">A505</font></span></a>, <a class="txt" href="#E11:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E66">A503</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E78">A533</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E79">A534</font></span></a>, <a class="ref" href="goboard7.html#T15" title="GOBOARD7:th.15">GOBOARD7:15</a></span>;<br/><a NAME="E6:35_1_1_13_1_2_1_1_2_1_1_4"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E57">A494</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="txt" href="#E7:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E62">A499</font></span></a>, <a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_2"><span class="lab"><font color="Green" title="E77">A523</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E78">A533</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:35_1_1_13_1_2_1_1_2_1_1_4"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_13"><span class="lab"><font color="Green" title="E55">A492</font></span></a>, <a class="txt" href="#E2:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E58">A495</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E5:35_1_1_13_1_2_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E80">A535</font></span></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_13_1_2_1_1_2"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_13_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_13_1_2_1_1_3"/><span class="lab"><font color="Green" title="E77">A536</font></span>: 
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_13_1_2_1_1_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_13_1_2_1_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_13_1_2_1_1_3_1_1"/>
( ( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> ) or ( <font color="Maroon" title="c12">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> ) or ( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> ) or ( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E65">A502</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_13_1_2_1_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_13_1_2_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E78">A537</font></span>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:35_1_1_13_1_2_1_1_3_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E57">A494</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="txt" href="#E7:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E62">A499</font></span></a>, <a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_3"><span class="lab"><font color="Green" title="E77">A536</font></span></a></span>;<br/><a NAME="E3:35_1_1_13_1_2_1_1_3_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E4:35_1_1_13_1_2_1_1_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A538</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E17:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E72">A509</font></span></a>, <a class="txt" href="#E14:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E69">A506</font></span></a>, <a class="txt" href="#E18:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E73">A510</font></span></a>, <a class="txt" href="#E16:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E71">A508</font></span></a>, <a class="txt" href="#E11:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E66">A503</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E78">A537</font></span></a>, <a class="ref" href="goboard7.html#T18" title="GOBOARD7:th.18">GOBOARD7:18</a></span>;<br/><a NAME="E5:35_1_1_13_1_2_1_1_3_1_1_1"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E57">A494</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="txt" href="#E7:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E62">A499</font></span></a>, <a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_3"><span class="lab"><font color="Green" title="E77">A536</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E78">A537</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_13_1_2_1_1_3_1_1_1"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_13"><span class="lab"><font color="Green" title="E55">A492</font></span></a>, <a class="txt" href="#E2:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E58">A495</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E4:35_1_1_13_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E79">A538</font></span></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_13_1_2_1_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_13_1_2_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E78">A539</font></span>: 
( <font color="Maroon" title="c12">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:35_1_1_13_1_2_1_1_3_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E57">A494</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="txt" href="#E7:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E62">A499</font></span></a>, <a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_3"><span class="lab"><font color="Green" title="E77">A536</font></span></a></span>;<br/><a NAME="E3:35_1_1_13_1_2_1_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E4:35_1_1_13_1_2_1_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A540</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E19:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E74">A511</font></span></a>, <a class="txt" href="#E14:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E69">A506</font></span></a>, <a class="txt" href="#E21:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E76">A513</font></span></a>, <a class="txt" href="#E11:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E66">A503</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E78">A539</font></span></a>, <a class="ref" href="goboard7.html#T20" title="GOBOARD7:th.20">GOBOARD7:20</a></span>;<br/><a NAME="E5:35_1_1_13_1_2_1_1_3_1_1_2"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E57">A494</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="txt" href="#E7:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E62">A499</font></span></a>, <a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_3"><span class="lab"><font color="Green" title="E77">A536</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E78">A539</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_13_1_2_1_1_3_1_1_2"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_13"><span class="lab"><font color="Green" title="E55">A492</font></span></a>, <a class="txt" href="#E2:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E58">A495</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E4:35_1_1_13_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E79">A540</font></span></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_13_1_2_1_1_3_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_13_1_2_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E78">A541</font></span>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:35_1_1_13_1_2_1_1_3_1_1_3"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E57">A494</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="txt" href="#E7:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E62">A499</font></span></a>, <a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_3"><span class="lab"><font color="Green" title="E77">A536</font></span></a></span>;<br/><a NAME="E3:35_1_1_13_1_2_1_1_3_1_1_3"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E4:35_1_1_13_1_2_1_1_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A542</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E17:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E72">A509</font></span></a>, <a class="txt" href="#E20:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E75">A512</font></span></a>, <a class="txt" href="#E15:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E70">A507</font></span></a>, <a class="txt" href="#E13:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E68">A505</font></span></a>, <a class="txt" href="#E11:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E66">A503</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E78">A541</font></span></a>, <a class="ref" href="goboard7.html#T17" title="GOBOARD7:th.17">GOBOARD7:17</a></span>;<br/><a NAME="E5:35_1_1_13_1_2_1_1_3_1_1_3"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E57">A494</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="txt" href="#E7:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E62">A499</font></span></a>, <a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_3"><span class="lab"><font color="Green" title="E77">A536</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E78">A541</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_13_1_2_1_1_3_1_1_3"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_13"><span class="lab"><font color="Green" title="E55">A492</font></span></a>, <a class="txt" href="#E2:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E58">A495</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E4:35_1_1_13_1_2_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E79">A542</font></span></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_13_1_2_1_1_3_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_13_1_2_1_1_3_1_1_4"/><span class="lab"><font color="Green" title="E78">A543</font></span>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:35_1_1_13_1_2_1_1_3_1_1_4"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E57">A494</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="txt" href="#E7:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E62">A499</font></span></a>, <a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_3"><span class="lab"><font color="Green" title="E77">A536</font></span></a></span>;<br/><a NAME="E3:35_1_1_13_1_2_1_1_3_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E4:35_1_1_13_1_2_1_1_3_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A544</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E19:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E74">A511</font></span></a>, <a class="txt" href="#E20:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E75">A512</font></span></a>, <a class="txt" href="#E12:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E67">A504</font></span></a>, <a class="txt" href="#E11:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E66">A503</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E78">A543</font></span></a>, <a class="ref" href="goboard7.html#T19" title="GOBOARD7:th.19">GOBOARD7:19</a></span>;<br/><a NAME="E5:35_1_1_13_1_2_1_1_3_1_1_4"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E48">A429</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E57">A494</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E6:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E61">A498</font></span></a>, <a class="txt" href="#E7:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E62">A499</font></span></a>, <a class="txt" href="#E8:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E63">A500</font></span></a>, <a class="txt" href="#E9:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E64">A501</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_3"><span class="lab"><font color="Green" title="E77">A536</font></span></a>, <a class="txt" href="#E1:35_1_1_13_1_2_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E78">A543</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_13_1_2_1_1_3_1_1_4"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_13"><span class="lab"><font color="Green" title="E55">A492</font></span></a>, <a class="txt" href="#E2:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E58">A495</font></span></a>, <a class="txt" href="#E3:35_1_1_13_1_2"><span class="lab"><font color="Green" title="E59">A496</font></span></a>, <a class="txt" href="#E4:35_1_1_13_1_2_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E79">A544</font></span></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_13_1_2_1_1_3"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E23:35_1_1_13_1_2"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_14"><span class="kw">now </span></a><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>8</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> $1 is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> ;<br/><span class="kw">assume </span><a NAME="E1:35_1_1_14"/><span class="lab"><font color="Green" title="E55">A545</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/>for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_14"/><span class="lab"><font color="Green" title="E56">A546</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>8</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>8</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_14_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c10">k</font>] implies <font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1_1_14_1"/><span class="lab"><font color="Green" title="E57">A547</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_14_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">n</font>, <font color="Maroon" title="c12">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c12">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">m</font> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_1_14_1_1"/><span class="lab"><font color="Green" title="E58">A548</font></span>: 
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_14_1_1"/><span class="lab"><font color="Green" title="E59">A549</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:35_1_1_14_1_1"/><span class="lab"><font color="Green" title="E60">A550</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><a NAME="E4:35_1_1_14_1_1"/><span class="lab"><font color="Green" title="E61">A551</font></span>: 
( <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_14_1_1"><span class="lab"><font color="Green" title="E58">A548</font></span></a>, <a class="txt" href="#E2:35_1_1_14_1_1"><span class="lab"><font color="Green" title="E59">A549</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E5:35_1_1_14_1_1"/><span class="lab"><font color="Green" title="E62">A552</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_14_1_1"><span class="lab"><font color="Green" title="E59">A549</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E6:35_1_1_14_1_1"/><span class="lab"><font color="Green" title="E63">A553</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_14_1_1"><span class="lab"><font color="Green" title="E58">A548</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E7:35_1_1_14_1_1"/><span class="lab"><font color="Green" title="E64">A554</font></span>: 
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:35_1_1"><span class="lab"><font color="Green" title="E51">A464</font></span></a></span>;<br/><a NAME="E8:35_1_1_14_1_1"/><span class="lab"><font color="Green" title="E65">A555</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a></span>;<br/><a NAME="E9:35_1_1_14_1_1"/><span class="lab"><font color="Green" title="E66">A556</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_14_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_14_1_1_1"/>
( ( <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> ) or ( <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> ) or ( <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or ( <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_14_1_1"><span class="lab"><font color="Green" title="E61">A551</font></span></a>, <a class="txt" href="#E9:35_1_1_14_1_1"><span class="lab"><font color="Green" title="E66">A556</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_14_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_14_1_1_1_1"/><span class="lab"><font color="Green" title="E67">A557</font></span>: 
( <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><a NAME="E2:35_1_1_14_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A558</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_14_1_1"><span class="lab"><font color="Green" title="E62">A552</font></span></a>, <a class="txt" href="#E8:35_1_1_14_1_1"><span class="lab"><font color="Green" title="E65">A555</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E3:35_1_1_14_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A559</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_14_1_1"><span class="lab"><font color="Green" title="E64">A554</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/><a NAME="E4:35_1_1_14_1_1_1_1"/><span class="lab"><font color="Green" title="E70">A560</font></span>: 
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_14_1_1"><span class="lab"><font color="Green" title="E63">A553</font></span></a>, <a class="txt" href="#E8:35_1_1_14_1_1"><span class="lab"><font color="Green" title="E65">A555</font></span></a>, <a class="txt" href="#E1:35_1_1_14_1_1_1_1"><span class="lab"><font color="Green" title="E67">A557</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E5:35_1_1_14_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_14_1_1"><span class="lab"><font color="Green" title="E64">A554</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_14_1_1_1_1"/>
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_14_1"><span class="lab"><font color="Green" title="E57">A547</font></span></a>, <a class="txt" href="#E3:35_1_1_14_1_1"><span class="lab"><font color="Green" title="E60">A550</font></span></a>, <a class="txt" href="#E4:35_1_1_14_1_1_1_1"><span class="lab"><font color="Green" title="E70">A560</font></span></a>, <a class="txt" href="#E2:35_1_1_14_1_1_1_1"><span class="lab"><font color="Green" title="E68">A558</font></span></a>, <a class="txt" href="#E3:35_1_1_14_1_1_1_1"><span class="lab"><font color="Green" title="E69">A559</font></span></a>, <a class="ref" href="partfun2.html#T17" title="PARTFUN2:th.17">PARTFUN2:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_14_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_14_1_1_1_2"/>
( <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_14_1_1_1_2"/>
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_14"><span class="lab"><font color="Green" title="E55">A545</font></span></a>, <a class="txt" href="#E2:35_1_1_14_1_1"><span class="lab"><font color="Green" title="E59">A549</font></span></a>, <a class="txt" href="#E3:35_1_1_14_1_1"><span class="lab"><font color="Green" title="E60">A550</font></span></a>, <a class="txt" href="#E9:35_1_1_14_1_1"><span class="lab"><font color="Green" title="E66">A556</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_14_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_14_1_1_1_3"/>
( <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_14_1_1_1_3"/>
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_14"><span class="lab"><font color="Green" title="E55">A545</font></span></a>, <a class="txt" href="#E1:35_1_1_14_1_1"><span class="lab"><font color="Green" title="E58">A548</font></span></a>, <a class="txt" href="#E3:35_1_1_14_1_1"><span class="lab"><font color="Green" title="E60">A550</font></span></a>, <a class="txt" href="#E9:35_1_1_14_1_1"><span class="lab"><font color="Green" title="E66">A556</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_14_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_14_1_1_1_4"/>
( <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_14_1_1_1_4"/>
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:35_1_1_14_1"/>
<font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T16" title="PARTFUN2:th.16">PARTFUN2:16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:35_1_1_14"/><span class="lab"><font color="Green" title="E57">A561</font></span>: 
<font color="Maroon">S<sub>8</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1"><span class="lab"><font color="Green" title="E34">A170</font></span></a></span>;<br/><a NAME="E4:35_1_1_14"/><span class="lab"><font color="Green" title="E58">A562</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>8</sub></font>[<font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E3:35_1_1_14"><span class="lab"><font color="Green" title="E57">A561</font></span></a>, <a class="txt" href="#E2:35_1_1_14"><span class="lab"><font color="Green" title="E56">A546</font></span></a>);<br/></span><a NAME="E5:35_1_1_14"/><span class="lab"><font color="Green" title="E59">A563</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_14_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font></span><br/>

<a NAME="E1:35_1_1_14_2"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_14"><span class="lab"><font color="Green" title="E58">A562</font></span></a></span>;<br/>
<span class="kw">hence </span> <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T77" title="FINSEQ_4:th.77">FINSEQ_4:77</a></span>
<br/>.= 
<font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">set </span><font color="Maroon" title="c10">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/><a NAME="E6:35_1_1_14"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a></span>;<br/><a NAME="E7:35_1_1_14"/><span class="kw">then </span>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd13.html#T8" title="GOBRD13:th.8">GOBRD13:8</a>, <a class="ref" href="gobrd13.html#T9" title="GOBRD13:th.9">GOBRD13:9</a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/><a NAME="E8:35_1_1_14"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E9:35_1_1_14"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_14"><span class="lab"><font color="Green" title="E59">A563</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E29:35_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A564</font></span>: 
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">k</font>]
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E31:35_1_1"/><span class="lab"><font color="Green" title="E56">A565</font></span>: 
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c10">k</font>]
 <span class="kw">and </span><br/><a NAME="E32:35_1_1"/><span class="lab"><font color="Green" title="E57">A566</font></span>: 
for <font color="Olive" title="b1">l</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">l</font>] holds <br/><font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E29:35_1_1"><span class="lab"><font color="Green" title="E55">A564</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c11">k</font> = <font color="Maroon" title="c10">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E35:35_1_1"/><span class="lab"><font color="Green" title="E58">A567</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c11">k</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E36:35_1_1"/><span class="lab"><font color="Green" title="E59">A568</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c11">k</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E37:35_1_1"/><span class="lab"><font color="Green" title="E60">A569</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c11">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:35_1_1"><span class="lab"><font color="Green" title="E56">A565</font></span></a></span>;<br/>
<a NAME="E38:35_1_1"/><span class="lab"><font color="Green" title="E61">A570</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:35_1_1"><span class="lab"><font color="Green" title="E58">A567</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">f</font> = <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c11">k</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:35_1_1"><span class="lab"><font color="Green" title="E56">A565</font></span></a></span>;<br/>
<a NAME="E40:35_1_1"/><span class="lab"><font color="Green" title="E62">A571</font></span>: 
<font color="Maroon" title="c13">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a></span>;<br/>
<a NAME="E41:35_1_1"/><span class="lab"><font color="Green" title="E63">A572</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:35_1_1"><span class="lab"><font color="Green" title="E58">A567</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E42:35_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A573</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:35_1_1"><span class="lab"><font color="Green" title="E59">A568</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E43:35_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:35_1_1"><span class="lab"><font color="Green" title="E61">A570</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E44:35_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A574</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E45:35_1_1"/><span class="lab"><font color="Green" title="E66">A575</font></span>: 
<font color="Maroon">S<sub>7</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1"><span class="lab"><font color="Green" title="E34">A170</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="sppol_2.html#T27" title="SPPOL_2:th.27">SPPOL_2:27</a></span>;<br/>
<a NAME="E46:35_1_1"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>7</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E45:35_1_1"><span class="lab"><font color="Green" title="E66">A575</font></span></a>, <a class="txt" href="#E27:35_1_1"><span class="lab"><font color="Green" title="E54">A491</font></span></a>);<br/></span>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">f</font> = <font color="Maroon" title="c13">f</font> as  non  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>   <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:35_1_1"><span class="lab"><font color="Green" title="E62">A571</font></span></a>, <a class="txt" href="#E44:35_1_1"><span class="lab"><font color="Green" title="E65">A574</font></span></a>, <a class="ref" href="jordan8.html#T7" title="JORDAN8:th.7">JORDAN8:7</a>, <a class="ref" href="jordan8.html#T8" title="JORDAN8:th.8">JORDAN8:8</a></span>;<br/>
<a NAME="E48:35_1_1"/><span class="lab"><font color="Green" title="E67">A576</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:35_1_1"><span class="lab"><font color="Green" title="E64">A573</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">g</font> = <font color="Maroon" title="c14">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>;<br/>
<a NAME="E49:35_1_1"/>
<font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>;<br/>
<a NAME="E50:35_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E51:35_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A577</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:35_1_1"><span class="lab"><font color="Green" title="E67">A576</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E52:35_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A578</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></span>;<br/>
<a NAME="E53:35_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:35_1_1"><span class="lab"><font color="Green" title="E68">A577</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">g</font> = <font color="Maroon" title="c14">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></span>;<br/>
<a NAME="E55:35_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E70">A579</font></span>: <font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font></span>)</span> = 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></span>
<br/>.= 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:35_1_1"><span class="lab"><font color="Green" title="E69">A578</font></span></a></span>
;<br/>
<a NAME="E57:35_1_1"/><span class="lab"><font color="Green" title="E71">A580</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a></span>;<br/>
<a NAME="E58:35_1_1"/><span class="lab"><font color="Green" title="E72">A581</font></span>: 
for <font color="Olive" title="b1">j</font>, <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font> &amp; <font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font> holds <br/><font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_16"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">j</font>, <font color="Maroon" title="c18">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">j</font> &amp; <font color="Maroon" title="c17">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font> &amp; <font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">j</font> implies <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_1_16"/><span class="lab"><font color="Green" title="E73">A582</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i</font>
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_16"/><span class="lab"><font color="Green" title="E74">A583</font></span>: 
<font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font>
 <span class="kw">and </span><br/><a NAME="E3:35_1_1_16"/><span class="lab"><font color="Green" title="E75">A584</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">j</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_16"/><span class="lab"><font color="Green" title="E76">A585</font></span>: 
<font color="Maroon" title="c17">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font>
 <span class="kw">and </span><br/><a NAME="E5:35_1_1_16"/><span class="lab"><font color="Green" title="E77">A586</font></span>: 
<font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">j</font>
 <span class="kw">and </span><br/><a NAME="E6:35_1_1_16"/><span class="lab"><font color="Green" title="E78">A587</font></span>: 
<font color="Maroon" title="c18">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E7:35_1_1_16"/><span class="lab"><font color="Green" title="E79">A588</font></span>: 
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_16"><span class="lab"><font color="Green" title="E73">A582</font></span></a>, <a class="txt" href="#E2:35_1_1_16"><span class="lab"><font color="Green" title="E74">A583</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E8:35_1_1_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A589</font></span>: 
<font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></span>;<br/>
<a NAME="E9:35_1_1_16"/><span class="lab"><font color="Green" title="E81">A590</font></span>: 
<font color="Maroon" title="c17">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_16"><span class="lab"><font color="Green" title="E75">A584</font></span></a>, <a class="txt" href="#E4:35_1_1_16"><span class="lab"><font color="Green" title="E76">A585</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E10:35_1_1_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A591</font></span>: 
<font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_16_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_16_1"/>
( <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">j</font> or <font color="Maroon" title="c17">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">i</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_16"><span class="lab"><font color="Green" title="E78">A587</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_16_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_16_1_1"/><span class="lab"><font color="Green" title="E83">A592</font></span>: 
<font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c19">l</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font>;<br/><span class="kw">set </span><font color="Maroon" title="c20">m9</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font>;<br/><a NAME="E2:35_1_1_16_1_1"/><span class="lab"><font color="Green" title="E84">A593</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_16_1_1"><span class="lab"><font color="Green" title="E83">A592</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E3:35_1_1_16_1_1"/><span class="lab"><font color="Green" title="E85">A594</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a></span>;<br/><a NAME="E4:35_1_1_16_1_1"/><span class="lab"><font color="Green" title="E86">A595</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E52:35_1_1"><span class="lab"><font color="Green" title="E69">A578</font></span></a>, <a class="txt" href="#E4:35_1_1_16"><span class="lab"><font color="Green" title="E76">A585</font></span></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></span>;<br/><a NAME="E5:35_1_1_16_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A596</font></span>: 
<font color="Maroon" title="c14">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:35_1_1"><span class="lab"><font color="Green" title="E53">A490</font></span></a></span>;<br/><a NAME="E6:35_1_1_16_1_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E7:35_1_1_16_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A597</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_16"><span class="lab"><font color="Green" title="E75">A584</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:35_1_1_16_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_16_1_1"><span class="lab"><font color="Green" title="E85">A594</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E9:35_1_1_16_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A598</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_16_1_1"><span class="lab"><font color="Green" title="E87">A596</font></span></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></span>;<br/><a NAME="E10:35_1_1_16_1_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E11:35_1_1_16_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_16"><span class="lab"><font color="Green" title="E73">A582</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:35_1_1_16_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A599</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_16_1_1"><span class="lab"><font color="Green" title="E84">A593</font></span></a>, <a class="txt" href="#E3:35_1_1_16_1_1"><span class="lab"><font color="Green" title="E85">A594</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E13:35_1_1_16_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_16_1_1"><span class="lab"><font color="Green" title="E87">A596</font></span></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></span>;<br/><span class="kw">hence </span><a NAME="E14:35_1_1_16_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:35_1_1"><span class="lab"><font color="Green" title="E57">A566</font></span></a>, <a class="txt" href="#E5:35_1_1_16"><span class="lab"><font color="Green" title="E77">A586</font></span></a>, <a class="txt" href="#E8:35_1_1_16"><span class="lab"><font color="Green" title="E80">A589</font></span></a>, <a class="txt" href="#E9:35_1_1_16"><span class="lab"><font color="Green" title="E81">A590</font></span></a>, <a class="txt" href="#E2:35_1_1_16_1_1"><span class="lab"><font color="Green" title="E84">A593</font></span></a>, <a class="txt" href="#E4:35_1_1_16_1_1"><span class="lab"><font color="Green" title="E86">A595</font></span></a>, <a class="txt" href="#E7:35_1_1_16_1_1"><span class="lab"><font color="Green" title="E88">A597</font></span></a>, <a class="txt" href="#E3:35_1_1_16_1_1"><span class="lab"><font color="Green" title="E85">A594</font></span></a>, <a class="txt" href="#E12:35_1_1_16_1_1"><span class="lab"><font color="Green" title="E90">A599</font></span></a>, <a class="txt" href="#E9:35_1_1_16_1_1"><span class="lab"><font color="Green" title="E89">A598</font></span></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_16_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_16_1_2"/><span class="lab"><font color="Green" title="E83">A600</font></span>: 
<font color="Maroon" title="c17">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c19">l</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font>;<br/><span class="kw">set </span><font color="Maroon" title="c20">m9</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font>;<br/><a NAME="E2:35_1_1_16_1_2"/><span class="lab"><font color="Green" title="E84">A601</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_16_1_2"><span class="lab"><font color="Green" title="E83">A600</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E3:35_1_1_16_1_2"/><span class="lab"><font color="Green" title="E85">A602</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a></span>;<br/><a NAME="E4:35_1_1_16_1_2"/><span class="lab"><font color="Green" title="E86">A603</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E52:35_1_1"><span class="lab"><font color="Green" title="E69">A578</font></span></a>, <a class="txt" href="#E2:35_1_1_16"><span class="lab"><font color="Green" title="E74">A583</font></span></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></span>;<br/><a NAME="E5:35_1_1_16_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A604</font></span>: 
<font color="Maroon" title="c14">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:35_1_1"><span class="lab"><font color="Green" title="E53">A490</font></span></a></span>;<br/><a NAME="E6:35_1_1_16_1_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E7:35_1_1_16_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A605</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_16"><span class="lab"><font color="Green" title="E73">A582</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:35_1_1_16_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_16_1_2"><span class="lab"><font color="Green" title="E85">A602</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E9:35_1_1_16_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A606</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_16_1_2"><span class="lab"><font color="Green" title="E87">A604</font></span></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></span>;<br/><a NAME="E10:35_1_1_16_1_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E11:35_1_1_16_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_16"><span class="lab"><font color="Green" title="E75">A584</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:35_1_1_16_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A607</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_16_1_2"><span class="lab"><font color="Green" title="E84">A601</font></span></a>, <a class="txt" href="#E3:35_1_1_16_1_2"><span class="lab"><font color="Green" title="E85">A602</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E13:35_1_1_16_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_16_1_2"><span class="lab"><font color="Green" title="E87">A604</font></span></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></span>;<br/><span class="kw">hence </span><a NAME="E14:35_1_1_16_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:35_1_1"><span class="lab"><font color="Green" title="E57">A566</font></span></a>, <a class="txt" href="#E5:35_1_1_16"><span class="lab"><font color="Green" title="E77">A586</font></span></a>, <a class="txt" href="#E7:35_1_1_16"><span class="lab"><font color="Green" title="E79">A588</font></span></a>, <a class="txt" href="#E10:35_1_1_16"><span class="lab"><font color="Green" title="E82">A591</font></span></a>, <a class="txt" href="#E2:35_1_1_16_1_2"><span class="lab"><font color="Green" title="E84">A601</font></span></a>, <a class="txt" href="#E4:35_1_1_16_1_2"><span class="lab"><font color="Green" title="E86">A603</font></span></a>, <a class="txt" href="#E7:35_1_1_16_1_2"><span class="lab"><font color="Green" title="E88">A605</font></span></a>, <a class="txt" href="#E3:35_1_1_16_1_2"><span class="lab"><font color="Green" title="E85">A602</font></span></a>, <a class="txt" href="#E12:35_1_1_16_1_2"><span class="lab"><font color="Green" title="E90">A607</font></span></a>, <a class="txt" href="#E9:35_1_1_16_1_2"><span class="lab"><font color="Green" title="E89">A606</font></span></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E73">A608</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_17"><span class="kw">now </span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c17">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E2:35_1_1_17"/><span class="lab"><font color="Green" title="E74">A609</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i</font> &amp; <font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E3:35_1_1_17"/><span class="lab"><font color="Green" title="E75">A610</font></span>: 
(  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c17">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c17">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_17"/><span class="lab"><font color="Green" title="E76">A611</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c17">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E35">A171</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c18">i</font> = <font color="Maroon" title="c17">i</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )</span><br/><span class="kw">thus </span><a NAME="E5:35_1_1_17"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_17"><span class="lab"><font color="Green" title="E74">A609</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )</span><br/><a NAME="E6:35_1_1_17"/><span class="lab"><font color="Green" title="E77">A612</font></span>: 
<font color="Maroon" title="c14">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:35_1_1"><span class="lab"><font color="Green" title="E53">A490</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E44:35_1_1"><span class="lab"><font color="Green" title="E65">A574</font></span></a></span>;<br/><a NAME="E7:35_1_1_17"/><span class="lab"><font color="Green" title="E78">A613</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:35_1_1"><span class="lab"><font color="Green" title="E65">A574</font></span></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></span>;<br/><a NAME="E8:35_1_1_17"/><span class="kw">then </span>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 2</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_17"><span class="lab"><font color="Green" title="E76">A611</font></span></a>, <a class="txt" href="#E6:35_1_1_17"><span class="lab"><font color="Green" title="E77">A612</font></span></a>, <a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )</span><br/><a NAME="E10:35_1_1_17"/>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_17"><span class="lab"><font color="Green" title="E78">A613</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 2</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_17"><span class="lab"><font color="Green" title="E76">A611</font></span></a>, <a class="txt" href="#E6:35_1_1_17"><span class="lab"><font color="Green" title="E77">A612</font></span></a>, <a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )</span><br/><span class="kw">thus </span><a NAME="E12:35_1_1_17"/>
(  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_17"><span class="lab"><font color="Green" title="E75">A610</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E60:35_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E74">A614</font></span>: <font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></span>
<br/>.= 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:35_1_1"><span class="lab"><font color="Green" title="E61">A570</font></span></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>
;<br/>
<a NAME="E62:35_1_1"/><span class="lab"><font color="Green" title="E75">A615</font></span>: 
for <font color="Olive" title="b1">j</font>, <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font> holds <br/><font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_19"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">j</font>, <font color="Maroon" title="c18">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">j</font> &amp; <font color="Maroon" title="c17">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font> implies <font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">j</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_1_19"/><span class="lab"><font color="Green" title="E76">A616</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">i</font>
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_19"/><span class="lab"><font color="Green" title="E77">A617</font></span>: 
<font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">j</font>
 <span class="kw">and </span><br/><a NAME="E3:35_1_1_19"/><span class="lab"><font color="Green" title="E78">A618</font></span>: 
<font color="Maroon" title="c17">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_19"/><span class="lab"><font color="Green" title="E79">A619</font></span>: 
<font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:35_1_1_19"/><span class="lab"><font color="Green" title="E80">A620</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_19"><span class="lab"><font color="Green" title="E76">A616</font></span></a>, <a class="txt" href="#E2:35_1_1_19"><span class="lab"><font color="Green" title="E77">A617</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E6:35_1_1_19"/><span class="lab"><font color="Green" title="E81">A621</font></span>: 
<font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_19"><span class="lab"><font color="Green" title="E77">A617</font></span></a>, <a class="txt" href="#E3:35_1_1_19"><span class="lab"><font color="Green" title="E78">A618</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:35_1_1_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A622</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_19"><span class="lab"><font color="Green" title="E76">A616</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_19_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_19_1"/>
( <font color="Maroon" title="c17">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font> or <font color="Maroon" title="c17">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_19_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_19_1_1"/>
<font color="Maroon" title="c17">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_19_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_19"><span class="lab"><font color="Green" title="E78">A618</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_19_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E1:35_1_1_19"><span class="lab"><font color="Green" title="E76">A616</font></span></a>, <a class="txt" href="#E2:35_1_1_19"><span class="lab"><font color="Green" title="E77">A617</font></span></a>, <a class="txt" href="#E4:35_1_1_19"><span class="lab"><font color="Green" title="E79">A619</font></span></a>, <a class="txt" href="#E5:35_1_1_19"><span class="lab"><font color="Green" title="E80">A620</font></span></a>, <a class="txt" href="#E6:35_1_1_19"><span class="lab"><font color="Green" title="E81">A621</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_19_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_19_1_2"/>
<font color="Maroon" title="c17">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:35_1_1_19_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:35_1_1"><span class="lab"><font color="Green" title="E60">A569</font></span></a>, <a class="txt" href="#E61:35_1_1"><span class="lab"><font color="Green" title="E74">A614</font></span></a>, <a class="txt" href="#E56:35_1_1"><span class="lab"><font color="Green" title="E70">A579</font></span></a>, <a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E1:35_1_1_19"><span class="lab"><font color="Green" title="E76">A616</font></span></a>, <a class="txt" href="#E2:35_1_1_19"><span class="lab"><font color="Green" title="E77">A617</font></span></a>, <a class="txt" href="#E4:35_1_1_19"><span class="lab"><font color="Green" title="E79">A619</font></span></a>, <a class="txt" href="#E7:35_1_1_19"><span class="lab"><font color="Green" title="E82">A622</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E63:35_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:35_1_1"><span class="lab"><font color="Green" title="E67">A576</font></span></a>, <a class="txt" href="#E52:35_1_1"><span class="lab"><font color="Green" title="E69">A578</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E64:35_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A623</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:35_1_1"><span class="lab"><font color="Green" title="E61">A570</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/>
<a NAME="E65:35_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A624</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font>
 
;<br/>
<a NAME="E66:35_1_1"/><span class="lab"><font color="Green" title="E78">A625</font></span>: 
<font color="Maroon" title="c16">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="ref" href="jordan8.html#T5" title="JORDAN8:th.5">JORDAN8:5</a></span>;<br/>
<a NAME="E67:35_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A626</font></span>: 
<font color="Maroon" title="c16">g</font> is  <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan8.html#T7" title="JORDAN8:th.7">JORDAN8:7</a></span>;<br/>
<a NAME="E68:35_1_1"/><span class="lab"><font color="Green" title="E80">A627</font></span>: 
not <font color="Maroon" title="c16">g</font> is  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_20"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
1
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="seqm_3.html#D15" title="SEQM_3:def.15">SEQM_3:def 15</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g</font> &amp; not <font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/>

<span class="kw">take </span>
2
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g</font> &amp; 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g</font> &amp; not <font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 )</span><br/>

<span class="kw">thus </span><a NAME="E1:35_1_1_20"/><span class="lab"><font color="Green" title="E81">A628</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g</font> &amp; not <font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 )</span><br/>

<span class="kw">thus </span><a NAME="E2:35_1_1_20"/><span class="lab"><font color="Green" title="E82">A629</font></span>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:35_1_1"><span class="lab"><font color="Green" title="E76">A623</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span><br/>

<a NAME="E3:35_1_1_20"/><span class="kw">then </span>
<font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E67:35_1_1"><span class="lab"><font color="Green" title="E79">A626</font></span></a>, <a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a>, <a class="ref" href="goboard7.html#T31" title="GOBOARD7:th.31">GOBOARD7:31</a></span>;<br/>
<a NAME="E4:35_1_1_20"/><span class="kw">then </span>
<font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_20"><span class="lab"><font color="Green" title="E81">A628</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:35_1_1_20"/>
not <font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_20"><span class="lab"><font color="Green" title="E82">A629</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E69:35_1_1"/><span class="lab"><font color="Green" title="E81">A630</font></span>: 
for <font color="Olive" title="b1">j</font>, <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font> holds <br/><font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_21"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">j</font>, <font color="Maroon" title="c18">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">j</font> &amp; <font color="Maroon" title="c17">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font> implies <font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">j</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_1_21"/><span class="lab"><font color="Green" title="E82">A631</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">j</font> &amp; <font color="Maroon" title="c17">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font> )
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_21"/><span class="lab"><font color="Green" title="E83">A632</font></span>: 
<font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:35_1_1_21"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">j</font> &amp; <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_21"><span class="lab"><font color="Green" title="E82">A631</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:35_1_1_21"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E1:35_1_1_21"><span class="lab"><font color="Green" title="E82">A631</font></span></a>, <a class="txt" href="#E2:35_1_1_21"><span class="lab"><font color="Green" title="E83">A632</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E70:35_1_1"/>
<font color="Maroon" title="c16">g</font> is  <a href="goboard5.html#V1" title="GOBOARD5:attr.1">s.c.c.</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_22"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="goboard5.html#D4" title="GOBOARD5:def.4">GOBOARD5:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or ( ( <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c17">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c16">g</font>,<font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c18">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or ( ( <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">j</font> ) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c17">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c18">j</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_1_22"/><span class="lab"><font color="Green" title="E82">A633</font></span>: 
<font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">j</font>
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_22"/><span class="lab"><font color="Green" title="E83">A634</font></span>: 
( ( <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c18">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font> ) or <font color="Maroon" title="c18">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c17">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c18">j</font></span><br/>

<a NAME="E3:35_1_1_22"/><span class="lab"><font color="Green" title="E84">A635</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E4:35_1_1_22"/><span class="lab"><font color="Green" title="E85">A636</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E5:35_1_1_22"/><span class="lab"><font color="Green" title="E86">A637</font></span>: 
<font color="Maroon" title="c18">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E6:35_1_1_22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A638</font></span>: 
<font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:35_1_1_22"/>
<font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E8:35_1_1_22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A639</font></span>: 
<font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_22"><span class="lab"><font color="Green" title="E86">A637</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_22_1"/>
( ( <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c18">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font> ) or ( <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c18">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font> ) or ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i</font> &amp; <font color="Maroon" title="c18">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_22"><span class="lab"><font color="Green" title="E83">A634</font></span></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E89">A640</font></span>: 
( <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c18">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c17">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c18">j</font></span><br/><div class="add"><a NAME="E2:35_1_1_22_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A641</font></span>: 
<font color="Maroon" title="c18">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:35_1_1_22_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A642</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c19">i19</font>, <font color="Maroon" title="c20">j19</font>, <font color="Maroon" title="c21">i29</font>, <font color="Maroon" title="c22">j29</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E92">A643</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E93">A644</font></span>: 
<font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i19</font>,<font color="Maroon" title="c20">j19</font>
 <span class="kw">and </span><br/><a NAME="E7:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E94">A645</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E95">A646</font></span>: 
<font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i29</font>,<font color="Maroon" title="c22">j29</font>
 <span class="kw">and </span><br/><a NAME="E9:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E96">A647</font></span>: 
( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:35_1_1"><span class="lab"><font color="Green" title="E78">A625</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E90">A641</font></span></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>;<br/><a NAME="E10:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E97">A648</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E92">A643</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E11:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E98">A649</font></span>: 
<font color="Maroon" title="c22">j29</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E94">A645</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E12:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E99">A650</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E94">A645</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E13:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E100">A651</font></span>: 
<font color="Maroon" title="c19">i19</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E92">A643</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E14:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E101">A652</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E94">A645</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E15:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E102">A653</font></span>: 
<font color="Maroon" title="c20">j19</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E92">A643</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E16:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E103">A654</font></span>: 
<font color="Maroon" title="c21">i29</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E94">A645</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E17:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E104">A655</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E92">A643</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E18:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E105">A656</font></span>: 
<font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E89">A640</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E19:35_1_1_22_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A657</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E89">A640</font></span></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></span>;<br/><a NAME="E20:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E107">A658</font></span>: 
<font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E105">A656</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c23">i1</font>, <font color="Maroon" title="c24">j1</font>, <font color="Maroon" title="c25">i2</font>, <font color="Maroon" title="c26">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E22:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E108">A659</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">i1</font>,<font color="Maroon" title="c24">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E23:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E109">A660</font></span>: 
<font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c23">i1</font>,<font color="Maroon" title="c24">j1</font>
 <span class="kw">and </span><br/><a NAME="E24:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E110">A661</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">i2</font>,<font color="Maroon" title="c26">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E25:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E111">A662</font></span>: 
<font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c25">i2</font>,<font color="Maroon" title="c26">j2</font>
 <span class="kw">and </span><br/><a NAME="E26:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E112">A663</font></span>: 
( ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:35_1_1"><span class="lab"><font color="Green" title="E78">A625</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E89">A640</font></span></a>, <a class="txt" href="#E18:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E105">A656</font></span></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>;<br/><a NAME="E27:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E113">A664</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E108">A659</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E28:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E114">A665</font></span>: 
<font color="Maroon" title="c26">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E110">A661</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E29:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E115">A666</font></span>: 
<font color="Maroon" title="c24">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E108">A659</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E30:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E116">A667</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E110">A661</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E31:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E117">A668</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E108">A659</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E32:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E118">A669</font></span>: 
<font color="Maroon" title="c25">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E110">A661</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E33:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E119">A670</font></span>: 
<font color="Maroon" title="c23">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E108">A659</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E34:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E120">A671</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E89">A640</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">assume </span><a NAME="E35:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E121">A672</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c17">i</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c18">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E36:35_1_1_22_1_1"/><span class="lab"><font color="Green" title="E122">A673</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E110">A661</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_22_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_22_1_1_1_1"/>
( ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> &amp; <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> &amp; <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> &amp; <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> &amp; <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or ( <font color="Maroon" title="c23">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> &amp; <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> &amp; <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> &amp; <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> &amp; <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> &amp; <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> &amp; <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> &amp; <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> &amp; <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E112">A663</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E96">A647</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_1_1_1_1"/><span class="lab"><font color="Green" title="E123">A674</font></span>: 
( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> &amp; <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A675</font></span>: 
( <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font> or <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c24">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E106">A657</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E27:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E113">A664</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E119">A670</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E117">A668</font></span></a>, <a class="txt" href="#E28:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E114">A665</font></span></a>, <a class="txt" href="#E3:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E91">A642</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E10:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E97">A648</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E100">A651</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E104">A655</font></span></a>, <a class="txt" href="#E11:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E98">A649</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E121">A672</font></span></a>, <a class="ref" href="goboard7.html#T24" title="GOBOARD7:th.24">GOBOARD7:24</a></span>;<br/><a NAME="E3:35_1_1_22_1_1_1_1_1"/>
<font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E106">A657</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E27:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E113">A664</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E119">A670</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E117">A668</font></span></a>, <a class="txt" href="#E28:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E114">A665</font></span></a>, <a class="txt" href="#E3:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E91">A642</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E10:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E97">A648</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E100">A651</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E104">A655</font></span></a>, <a class="txt" href="#E11:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E98">A649</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E121">A672</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1_1_1_1"><span class="lab"><font color="Green" title="E123">A674</font></span></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_22_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E62:35_1_1"><span class="lab"><font color="Green" title="E75">A615</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E5:35_1_1_22"><span class="lab"><font color="Green" title="E86">A637</font></span></a>, <a class="txt" href="#E4:35_1_1_22"><span class="lab"><font color="Green" title="E85">A636</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E8:35_1_1_22"><span class="lab"><font color="Green" title="E88">A639</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E89">A640</font></span></a>, <a class="txt" href="#E18:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E105">A656</font></span></a>, <a class="txt" href="#E20:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E107">A658</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E90">A641</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1_1_1_1"><span class="lab"><font color="Green" title="E123">A674</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_1_1_1_1"><span class="lab"><font color="Green" title="E124">A675</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_1_1_1_2"/><span class="lab"><font color="Green" title="E123">A676</font></span>: 
( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> &amp; <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_1_1_1_2"/><span class="kw">then </span>
( ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E106">A657</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E27:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E113">A664</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E119">A670</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E117">A668</font></span></a>, <a class="txt" href="#E28:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E114">A665</font></span></a>, <a class="txt" href="#E3:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E91">A642</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E10:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E97">A648</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E104">A655</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E102">A653</font></span></a>, <a class="txt" href="#E16:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E103">A654</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E121">A672</font></span></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_22_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E62:35_1_1"><span class="lab"><font color="Green" title="E75">A615</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E8:35_1_1_22"><span class="lab"><font color="Green" title="E88">A639</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E89">A640</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E120">A671</font></span></a>, <a class="txt" href="#E20:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E107">A658</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E90">A641</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1_1_1_2"><span class="lab"><font color="Green" title="E123">A676</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_1_1_1_3"/><span class="lab"><font color="Green" title="E123">A677</font></span>: 
( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> &amp; <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_1_1_1_3"/><span class="kw">then </span>
( ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j1</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j1</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E106">A657</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E27:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E113">A664</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E119">A670</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E117">A668</font></span></a>, <a class="txt" href="#E28:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E114">A665</font></span></a>, <a class="txt" href="#E3:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E91">A642</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E100">A651</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E104">A655</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E102">A653</font></span></a>, <a class="txt" href="#E12:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E99">A650</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E121">A672</font></span></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_22_1_1_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E62:35_1_1"><span class="lab"><font color="Green" title="E75">A615</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E8:35_1_1_22"><span class="lab"><font color="Green" title="E88">A639</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E89">A640</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E120">A671</font></span></a>, <a class="txt" href="#E20:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E107">A658</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E90">A641</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1_1_1_3"><span class="lab"><font color="Green" title="E123">A677</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_1_1_1_4"/><span class="lab"><font color="Green" title="E123">A678</font></span>: 
( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> &amp; <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A679</font></span>: 
( <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> or <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c24">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E106">A657</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E27:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E113">A664</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E119">A670</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E117">A668</font></span></a>, <a class="txt" href="#E28:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E114">A665</font></span></a>, <a class="txt" href="#E3:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E91">A642</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E10:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E97">A648</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E100">A651</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E102">A653</font></span></a>, <a class="txt" href="#E14:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E101">A652</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E121">A672</font></span></a>, <a class="ref" href="goboard7.html#T24" title="GOBOARD7:th.24">GOBOARD7:24</a></span>;<br/><a NAME="E3:35_1_1_22_1_1_1_1_4"/>
<font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E106">A657</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E27:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E113">A664</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E119">A670</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E117">A668</font></span></a>, <a class="txt" href="#E28:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E114">A665</font></span></a>, <a class="txt" href="#E3:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E91">A642</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E10:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E97">A648</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E100">A651</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E102">A653</font></span></a>, <a class="txt" href="#E14:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E101">A652</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E121">A672</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1_1_1_4"><span class="lab"><font color="Green" title="E123">A678</font></span></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_22_1_1_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E62:35_1_1"><span class="lab"><font color="Green" title="E75">A615</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E89">A640</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E120">A671</font></span></a>, <a class="txt" href="#E20:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E107">A658</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E90">A641</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1_1_1_4"><span class="lab"><font color="Green" title="E123">A678</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_1_1_1_4"><span class="lab"><font color="Green" title="E124">A679</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_1_1_1_5"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_1_1_1_5"/><span class="lab"><font color="Green" title="E123">A680</font></span>: 
( <font color="Maroon" title="c23">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> &amp; <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_1_1_1_5"/><span class="kw">then </span>
( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i1</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i1</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j1</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E106">A657</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E27:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E113">A664</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E117">A668</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E115">A666</font></span></a>, <a class="txt" href="#E32:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E118">A669</font></span></a>, <a class="txt" href="#E3:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E91">A642</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E10:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E97">A648</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E100">A651</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E104">A655</font></span></a>, <a class="txt" href="#E11:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E98">A649</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E121">A672</font></span></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_22_1_1_1_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E62:35_1_1"><span class="lab"><font color="Green" title="E75">A615</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E8:35_1_1_22"><span class="lab"><font color="Green" title="E88">A639</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E89">A640</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E120">A671</font></span></a>, <a class="txt" href="#E20:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E107">A658</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E90">A641</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1_1_1_5"><span class="lab"><font color="Green" title="E123">A680</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_1_1_1_6"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_1_1_1_6"/><span class="lab"><font color="Green" title="E123">A681</font></span>: 
( <font color="Maroon" title="c23">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> &amp; <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_1_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A682</font></span>: 
( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font> or <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c23">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E106">A657</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E27:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E113">A664</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E117">A668</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E115">A666</font></span></a>, <a class="txt" href="#E32:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E118">A669</font></span></a>, <a class="txt" href="#E3:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E91">A642</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E10:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E97">A648</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E104">A655</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E102">A653</font></span></a>, <a class="txt" href="#E16:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E103">A654</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E121">A672</font></span></a>, <a class="ref" href="goboard7.html#T25" title="GOBOARD7:th.25">GOBOARD7:25</a></span>;<br/><a NAME="E3:35_1_1_22_1_1_1_1_6"/>
<font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E106">A657</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E27:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E113">A664</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E117">A668</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E115">A666</font></span></a>, <a class="txt" href="#E32:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E118">A669</font></span></a>, <a class="txt" href="#E3:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E91">A642</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E10:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E97">A648</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E104">A655</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E102">A653</font></span></a>, <a class="txt" href="#E16:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E103">A654</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E121">A672</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1_1_1_6"><span class="lab"><font color="Green" title="E123">A681</font></span></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_22_1_1_1_1_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E62:35_1_1"><span class="lab"><font color="Green" title="E75">A615</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E5:35_1_1_22"><span class="lab"><font color="Green" title="E86">A637</font></span></a>, <a class="txt" href="#E4:35_1_1_22"><span class="lab"><font color="Green" title="E85">A636</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E8:35_1_1_22"><span class="lab"><font color="Green" title="E88">A639</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E89">A640</font></span></a>, <a class="txt" href="#E18:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E105">A656</font></span></a>, <a class="txt" href="#E20:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E107">A658</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E90">A641</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1_1_1_6"><span class="lab"><font color="Green" title="E123">A681</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_1_1_1_6"><span class="lab"><font color="Green" title="E124">A682</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_1_1_1_7"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_1_1_1_7"/><span class="lab"><font color="Green" title="E123">A683</font></span>: 
( <font color="Maroon" title="c23">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> &amp; <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A684</font></span>: 
( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> or <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c23">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E106">A657</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E27:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E113">A664</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E117">A668</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E115">A666</font></span></a>, <a class="txt" href="#E32:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E118">A669</font></span></a>, <a class="txt" href="#E3:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E91">A642</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E100">A651</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E104">A655</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E102">A653</font></span></a>, <a class="txt" href="#E12:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E99">A650</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E121">A672</font></span></a>, <a class="ref" href="goboard7.html#T25" title="GOBOARD7:th.25">GOBOARD7:25</a></span>;<br/><a NAME="E3:35_1_1_22_1_1_1_1_7"/>
<font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E106">A657</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E27:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E113">A664</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E117">A668</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E115">A666</font></span></a>, <a class="txt" href="#E32:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E118">A669</font></span></a>, <a class="txt" href="#E3:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E91">A642</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E100">A651</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E104">A655</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E102">A653</font></span></a>, <a class="txt" href="#E12:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E99">A650</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E121">A672</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1_1_1_7"><span class="lab"><font color="Green" title="E123">A683</font></span></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_22_1_1_1_1_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E62:35_1_1"><span class="lab"><font color="Green" title="E75">A615</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E8:35_1_1_22"><span class="lab"><font color="Green" title="E88">A639</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E89">A640</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E120">A671</font></span></a>, <a class="txt" href="#E20:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E107">A658</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E90">A641</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1_1_1_7"><span class="lab"><font color="Green" title="E123">A683</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_1_1_1_7"><span class="lab"><font color="Green" title="E124">A684</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_1_1_1_8"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_1_1_1_8"/><span class="lab"><font color="Green" title="E123">A685</font></span>: 
( <font color="Maroon" title="c23">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> &amp; <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_1_1_1_8"/><span class="kw">then </span>
( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i1</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i1</font> &amp; <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j1</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E106">A657</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E27:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E113">A664</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E117">A668</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E115">A666</font></span></a>, <a class="txt" href="#E32:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E118">A669</font></span></a>, <a class="txt" href="#E3:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E91">A642</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E10:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E97">A648</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E100">A651</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E102">A653</font></span></a>, <a class="txt" href="#E14:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E101">A652</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E121">A672</font></span></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_22_1_1_1_1_8"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E62:35_1_1"><span class="lab"><font color="Green" title="E75">A615</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E8:35_1_1_22"><span class="lab"><font color="Green" title="E88">A639</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E89">A640</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E120">A671</font></span></a>, <a class="txt" href="#E20:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E107">A658</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E90">A641</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1_1_1_8"><span class="lab"><font color="Green" title="E123">A685</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_1_1_1_9"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_1_1_1_9"/><span class="lab"><font color="Green" title="E123">A686</font></span>: 
( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> &amp; <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_1_1_1_9"/><span class="kw">then </span>
( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j1</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j1</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j1</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E106">A657</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E119">A670</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E117">A668</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E115">A666</font></span></a>, <a class="txt" href="#E36:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E122">A673</font></span></a>, <a class="txt" href="#E3:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E91">A642</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E10:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E97">A648</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E100">A651</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E104">A655</font></span></a>, <a class="txt" href="#E11:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E98">A649</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E121">A672</font></span></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_22_1_1_1_1_9"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E62:35_1_1"><span class="lab"><font color="Green" title="E75">A615</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E8:35_1_1_22"><span class="lab"><font color="Green" title="E88">A639</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E89">A640</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E120">A671</font></span></a>, <a class="txt" href="#E20:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E107">A658</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E90">A641</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1_1_1_9"><span class="lab"><font color="Green" title="E123">A686</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_1_1_1_10"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_1_1_1_10"/><span class="lab"><font color="Green" title="E123">A687</font></span>: 
( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> &amp; <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_1_1_1_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A688</font></span>: 
( <font color="Maroon" title="c25">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font> or <font color="Maroon" title="c25">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c25">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E106">A657</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E119">A670</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E117">A668</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E115">A666</font></span></a>, <a class="txt" href="#E36:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E122">A673</font></span></a>, <a class="txt" href="#E3:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E91">A642</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E10:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E97">A648</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E104">A655</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E102">A653</font></span></a>, <a class="txt" href="#E16:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E103">A654</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E121">A672</font></span></a>, <a class="ref" href="goboard7.html#T25" title="GOBOARD7:th.25">GOBOARD7:25</a></span>;<br/><a NAME="E3:35_1_1_22_1_1_1_1_10"/>
<font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E106">A657</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E119">A670</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E117">A668</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E115">A666</font></span></a>, <a class="txt" href="#E36:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E122">A673</font></span></a>, <a class="txt" href="#E3:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E91">A642</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E10:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E97">A648</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E104">A655</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E102">A653</font></span></a>, <a class="txt" href="#E16:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E103">A654</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E121">A672</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1_1_1_10"><span class="lab"><font color="Green" title="E123">A687</font></span></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_22_1_1_1_1_10"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E62:35_1_1"><span class="lab"><font color="Green" title="E75">A615</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E89">A640</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E120">A671</font></span></a>, <a class="txt" href="#E20:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E107">A658</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E90">A641</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1_1_1_10"><span class="lab"><font color="Green" title="E123">A687</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_1_1_1_10"><span class="lab"><font color="Green" title="E124">A688</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_1_1_1_11"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_1_1_1_11"/><span class="lab"><font color="Green" title="E123">A689</font></span>: 
( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> &amp; <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_1_1_1_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A690</font></span>: 
( <font color="Maroon" title="c25">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> or <font color="Maroon" title="c25">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c25">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E106">A657</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E119">A670</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E117">A668</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E115">A666</font></span></a>, <a class="txt" href="#E36:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E122">A673</font></span></a>, <a class="txt" href="#E3:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E91">A642</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E100">A651</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E104">A655</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E102">A653</font></span></a>, <a class="txt" href="#E12:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E99">A650</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E121">A672</font></span></a>, <a class="ref" href="goboard7.html#T25" title="GOBOARD7:th.25">GOBOARD7:25</a></span>;<br/><a NAME="E3:35_1_1_22_1_1_1_1_11"/>
<font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E106">A657</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E119">A670</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E117">A668</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E115">A666</font></span></a>, <a class="txt" href="#E36:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E122">A673</font></span></a>, <a class="txt" href="#E3:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E91">A642</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E100">A651</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E104">A655</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E102">A653</font></span></a>, <a class="txt" href="#E12:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E99">A650</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E121">A672</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1_1_1_11"><span class="lab"><font color="Green" title="E123">A689</font></span></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_22_1_1_1_1_11"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E62:35_1_1"><span class="lab"><font color="Green" title="E75">A615</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E8:35_1_1_22"><span class="lab"><font color="Green" title="E88">A639</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E89">A640</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E120">A671</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E90">A641</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1_1_1_11"><span class="lab"><font color="Green" title="E123">A689</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_1_1_1_11"><span class="lab"><font color="Green" title="E124">A690</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_1_1_1_12"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_1_1_1_12"/><span class="lab"><font color="Green" title="E123">A691</font></span>: 
( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> &amp; <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_1_1_1_12"/><span class="kw">then </span>
( ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c22">j29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j1</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j1</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j1</font> ) or ( <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E106">A657</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E119">A670</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E117">A668</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E115">A666</font></span></a>, <a class="txt" href="#E36:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E122">A673</font></span></a>, <a class="txt" href="#E3:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E91">A642</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E10:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E97">A648</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E100">A651</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E102">A653</font></span></a>, <a class="txt" href="#E14:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E101">A652</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E121">A672</font></span></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_22_1_1_1_1_12"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E62:35_1_1"><span class="lab"><font color="Green" title="E75">A615</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E8:35_1_1_22"><span class="lab"><font color="Green" title="E88">A639</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E89">A640</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E120">A671</font></span></a>, <a class="txt" href="#E20:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E107">A658</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E90">A641</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1_1_1_12"><span class="lab"><font color="Green" title="E123">A691</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_1_1_1_13"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_1_1_1_13"/><span class="lab"><font color="Green" title="E123">A692</font></span>: 
( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_1_1_1_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A693</font></span>: 
( <font color="Maroon" title="c26">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font> or <font color="Maroon" title="c26">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c26">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E106">A657</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E27:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E113">A664</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E119">A670</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E115">A666</font></span></a>, <a class="txt" href="#E30:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E116">A667</font></span></a>, <a class="txt" href="#E3:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E91">A642</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E10:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E97">A648</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E100">A651</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E104">A655</font></span></a>, <a class="txt" href="#E11:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E98">A649</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E121">A672</font></span></a>, <a class="ref" href="goboard7.html#T24" title="GOBOARD7:th.24">GOBOARD7:24</a></span>;<br/><a NAME="E3:35_1_1_22_1_1_1_1_13"/>
<font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E106">A657</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E27:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E113">A664</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E119">A670</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E115">A666</font></span></a>, <a class="txt" href="#E30:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E116">A667</font></span></a>, <a class="txt" href="#E3:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E91">A642</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E10:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E97">A648</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E100">A651</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E104">A655</font></span></a>, <a class="txt" href="#E11:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E98">A649</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E121">A672</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1_1_1_13"><span class="lab"><font color="Green" title="E123">A692</font></span></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_22_1_1_1_1_13"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E62:35_1_1"><span class="lab"><font color="Green" title="E75">A615</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E89">A640</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E120">A671</font></span></a>, <a class="txt" href="#E20:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E107">A658</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E90">A641</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1_1_1_13"><span class="lab"><font color="Green" title="E123">A692</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_1_1_1_13"><span class="lab"><font color="Green" title="E124">A693</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_1_1_1_14"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_1_1_1_14"/><span class="lab"><font color="Green" title="E123">A694</font></span>: 
( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_1_1_1_14"/><span class="kw">then </span>
( ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font> &amp; <font color="Maroon" title="c26">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font> &amp; <font color="Maroon" title="c26">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c26">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c26">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E106">A657</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E27:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E113">A664</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E119">A670</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E115">A666</font></span></a>, <a class="txt" href="#E30:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E116">A667</font></span></a>, <a class="txt" href="#E3:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E91">A642</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E10:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E97">A648</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E104">A655</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E102">A653</font></span></a>, <a class="txt" href="#E16:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E103">A654</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E121">A672</font></span></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_22_1_1_1_1_14"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E62:35_1_1"><span class="lab"><font color="Green" title="E75">A615</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E8:35_1_1_22"><span class="lab"><font color="Green" title="E88">A639</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E89">A640</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E120">A671</font></span></a>, <a class="txt" href="#E20:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E107">A658</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E90">A641</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1_1_1_14"><span class="lab"><font color="Green" title="E123">A694</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_1_1_1_15"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_1_1_1_15"/><span class="lab"><font color="Green" title="E123">A695</font></span>: 
( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_1_1_1_15"/><span class="kw">then </span>
( ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c26">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c26">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c26">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font> ) or ( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c26">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j19</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E106">A657</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E27:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E113">A664</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E119">A670</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E115">A666</font></span></a>, <a class="txt" href="#E30:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E116">A667</font></span></a>, <a class="txt" href="#E3:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E91">A642</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E100">A651</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E104">A655</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E102">A653</font></span></a>, <a class="txt" href="#E12:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E99">A650</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E121">A672</font></span></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_22_1_1_1_1_15"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E62:35_1_1"><span class="lab"><font color="Green" title="E75">A615</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E8:35_1_1_22"><span class="lab"><font color="Green" title="E88">A639</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E89">A640</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E120">A671</font></span></a>, <a class="txt" href="#E20:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E107">A658</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E90">A641</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1_1_1_15"><span class="lab"><font color="Green" title="E123">A695</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_1_1_1_16"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_1_1_1_16"/><span class="lab"><font color="Green" title="E123">A696</font></span>: 
( <font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i2</font> &amp; <font color="Maroon" title="c24">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c19">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i29</font> &amp; <font color="Maroon" title="c20">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_1_1_1_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A697</font></span>: 
( <font color="Maroon" title="c26">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> or <font color="Maroon" title="c26">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c26">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E106">A657</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E27:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E113">A664</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E119">A670</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E115">A666</font></span></a>, <a class="txt" href="#E30:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E116">A667</font></span></a>, <a class="txt" href="#E3:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E91">A642</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E10:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E97">A648</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E100">A651</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E102">A653</font></span></a>, <a class="txt" href="#E14:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E101">A652</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E121">A672</font></span></a>, <a class="ref" href="goboard7.html#T24" title="GOBOARD7:th.24">GOBOARD7:24</a></span>;<br/><a NAME="E3:35_1_1_22_1_1_1_1_16"/>
<font color="Maroon" title="c23">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E106">A657</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E27:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E113">A664</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E119">A670</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E115">A666</font></span></a>, <a class="txt" href="#E30:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E116">A667</font></span></a>, <a class="txt" href="#E3:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E91">A642</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E10:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E97">A648</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E100">A651</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E102">A653</font></span></a>, <a class="txt" href="#E14:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E101">A652</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E121">A672</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1_1_1_16"><span class="lab"><font color="Green" title="E123">A696</font></span></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_22_1_1_1_1_16"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E62:35_1_1"><span class="lab"><font color="Green" title="E75">A615</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E8:35_1_1_22"><span class="lab"><font color="Green" title="E88">A639</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E89">A640</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E120">A671</font></span></a>, <a class="txt" href="#E23:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E109">A660</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E111">A662</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E90">A641</font></span></a>, <a class="txt" href="#E6:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E93">A644</font></span></a>, <a class="txt" href="#E8:35_1_1_22_1_1"><span class="lab"><font color="Green" title="E95">A646</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_1_1_1_16"><span class="lab"><font color="Green" title="E123">A696</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_1_1_1_16"><span class="lab"><font color="Green" title="E124">A697</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E38:35_1_1_22_1_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_2"/>
( <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c18">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c17">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c18">j</font></span><br/><div class="add"><a NAME="E2:35_1_1_22_1_2"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_22_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c17">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c18">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T65" title="XBOOLE_1:th.65">XBOOLE_1:65</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E89">A698</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i</font> &amp; <font color="Maroon" title="c18">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c17">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c18">j</font></span><br/><div class="add"><a NAME="E2:35_1_1_22_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A699</font></span>: 
<font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:35_1_1_22_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A700</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E89">A698</font></span></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></span>;<br/><a NAME="E4:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E92">A701</font></span>: 
<font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E90">A699</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c19">i1</font>, <font color="Maroon" title="c20">j1</font>, <font color="Maroon" title="c21">i2</font>, <font color="Maroon" title="c22">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E6:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E93">A702</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i1</font>,<font color="Maroon" title="c20">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E94">A703</font></span>: 
<font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<font color="Maroon" title="c20">j1</font>
 <span class="kw">and </span><br/><a NAME="E8:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E95">A704</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i2</font>,<font color="Maroon" title="c22">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E96">A705</font></span>: 
<font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i2</font>,<font color="Maroon" title="c22">j2</font>
 <span class="kw">and </span><br/><a NAME="E10:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E97">A706</font></span>: 
( ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:35_1_1"><span class="lab"><font color="Green" title="E78">A625</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E89">A698</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E90">A699</font></span></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>;<br/><a NAME="E11:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E98">A707</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E93">A702</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E12:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E99">A708</font></span>: 
<font color="Maroon" title="c22">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E95">A704</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E13:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E100">A709</font></span>: 
<font color="Maroon" title="c20">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E93">A702</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E14:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E101">A710</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E95">A704</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E15:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E102">A711</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E93">A702</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E16:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E103">A712</font></span>: 
<font color="Maroon" title="c21">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E95">A704</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E17:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E104">A713</font></span>: 
<font color="Maroon" title="c19">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E93">A702</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E18:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E105">A714</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E89">A698</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">assume </span><a NAME="E19:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E106">A715</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c17">i</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c18">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">consider </span><font color="Maroon" title="c23">i19</font>, <font color="Maroon" title="c24">j19</font>, <font color="Maroon" title="c25">i29</font>, <font color="Maroon" title="c26">j29</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E21:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E107">A716</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">i19</font>,<font color="Maroon" title="c24">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E22:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E108">A717</font></span>: 
<font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c23">i19</font>,<font color="Maroon" title="c24">j19</font>
 <span class="kw">and </span><br/><a NAME="E23:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E109">A718</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">i29</font>,<font color="Maroon" title="c26">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E24:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E110">A719</font></span>: 
<font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c25">i29</font>,<font color="Maroon" title="c26">j29</font>
 <span class="kw">and </span><br/><a NAME="E25:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E111">A720</font></span>: 
( ( <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> ) or ( <font color="Maroon" title="c23">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> ) or ( <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> ) or ( <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:35_1_1"><span class="lab"><font color="Green" title="E78">A625</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E89">A698</font></span></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>;<br/><a NAME="E26:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E112">A721</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E107">A716</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E27:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E113">A722</font></span>: 
<font color="Maroon" title="c26">j29</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E109">A718</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E28:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E114">A723</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E109">A718</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E29:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E115">A724</font></span>: 
<font color="Maroon" title="c23">i19</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E107">A716</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E30:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E116">A725</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E109">A718</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E31:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E117">A726</font></span>: 
<font color="Maroon" title="c24">j19</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E107">A716</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E32:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E118">A727</font></span>: 
<font color="Maroon" title="c25">i29</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E109">A718</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E33:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E119">A728</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E107">A716</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E34:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E120">A729</font></span>: 
<font color="Maroon" title="c18">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E89">A698</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E35:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E121">A730</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E89">A698</font></span></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></span>;<br/><a NAME="E36:35_1_1_22_1_3"/><span class="lab"><font color="Green" title="E122">A731</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E95">A704</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_22_1_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_22_1_3_1_1"/>
( ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> &amp; <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> &amp; <font color="Maroon" title="c23">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> &amp; <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> &amp; <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or ( <font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> &amp; <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> &amp; <font color="Maroon" title="c23">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> &amp; <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> &amp; <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> &amp; <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> &amp; <font color="Maroon" title="c23">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> &amp; <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> &amp; <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c23">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E97">A706</font></span></a>, <a class="txt" href="#E25:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E111">A720</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_3_1_1_1"/><span class="lab"><font color="Green" title="E123">A732</font></span>: 
( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> &amp; <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A733</font></span>: 
( <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j19</font> or <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j19</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E91">A700</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E11:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E98">A707</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E104">A713</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E102">A711</font></span></a>, <a class="txt" href="#E12:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E99">A708</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E121">A730</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E26:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E112">A721</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E115">A724</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E119">A728</font></span></a>, <a class="txt" href="#E27:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E113">A722</font></span></a>, <a class="txt" href="#E19:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E106">A715</font></span></a>, <a class="ref" href="goboard7.html#T24" title="GOBOARD7:th.24">GOBOARD7:24</a></span>;<br/><a NAME="E3:35_1_1_22_1_3_1_1_1"/>
<font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E91">A700</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E11:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E98">A707</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E104">A713</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E102">A711</font></span></a>, <a class="txt" href="#E12:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E99">A708</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E121">A730</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E26:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E112">A721</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E115">A724</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E119">A728</font></span></a>, <a class="txt" href="#E27:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E113">A722</font></span></a>, <a class="txt" href="#E19:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E106">A715</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3_1_1_1"><span class="lab"><font color="Green" title="E123">A732</font></span></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_22_1_3_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:35_1_1"><span class="lab"><font color="Green" title="E81">A630</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E4:35_1_1_22"><span class="lab"><font color="Green" title="E85">A636</font></span></a>, <a class="txt" href="#E8:35_1_1_22"><span class="lab"><font color="Green" title="E88">A639</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E89">A698</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E120">A729</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3_1_1_1"><span class="lab"><font color="Green" title="E123">A732</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_3_1_1_1"><span class="lab"><font color="Green" title="E124">A733</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_3_1_1_2"/><span class="lab"><font color="Green" title="E123">A734</font></span>: 
( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> &amp; <font color="Maroon" title="c23">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_3_1_1_2"/><span class="kw">then </span>
( ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i19</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j19</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i19</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j19</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j19</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j19</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E91">A700</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E11:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E98">A707</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E104">A713</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E102">A711</font></span></a>, <a class="txt" href="#E12:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E99">A708</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E121">A730</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E26:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E112">A721</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E119">A728</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E117">A726</font></span></a>, <a class="txt" href="#E32:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E118">A727</font></span></a>, <a class="txt" href="#E19:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E106">A715</font></span></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_22_1_3_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E69:35_1_1"><span class="lab"><font color="Green" title="E81">A630</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E8:35_1_1_22"><span class="lab"><font color="Green" title="E88">A639</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E89">A698</font></span></a>, <a class="txt" href="#E18:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E105">A714</font></span></a>, <a class="txt" href="#E4:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E92">A701</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E120">A729</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3_1_1_2"><span class="lab"><font color="Green" title="E123">A734</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_3_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_3_1_1_3"/><span class="lab"><font color="Green" title="E123">A735</font></span>: 
( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> &amp; <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_3_1_1_3"/><span class="kw">then </span>
( ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j19</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j19</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E91">A700</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E11:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E98">A707</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E104">A713</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E102">A711</font></span></a>, <a class="txt" href="#E12:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E99">A708</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E121">A730</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E115">A724</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E119">A728</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E117">A726</font></span></a>, <a class="txt" href="#E28:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E114">A723</font></span></a>, <a class="txt" href="#E19:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E106">A715</font></span></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_22_1_3_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E69:35_1_1"><span class="lab"><font color="Green" title="E81">A630</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E8:35_1_1_22"><span class="lab"><font color="Green" title="E88">A639</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E89">A698</font></span></a>, <a class="txt" href="#E18:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E105">A714</font></span></a>, <a class="txt" href="#E4:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E92">A701</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E120">A729</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3_1_1_3"><span class="lab"><font color="Green" title="E123">A735</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_3_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_3_1_1_4"/><span class="lab"><font color="Green" title="E123">A736</font></span>: 
( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> &amp; <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_3_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A737</font></span>: 
( <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> or <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E91">A700</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E11:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E98">A707</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E104">A713</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E102">A711</font></span></a>, <a class="txt" href="#E12:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E99">A708</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E121">A730</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E26:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E112">A721</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E115">A724</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E117">A726</font></span></a>, <a class="txt" href="#E30:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E116">A725</font></span></a>, <a class="txt" href="#E19:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E106">A715</font></span></a>, <a class="ref" href="goboard7.html#T24" title="GOBOARD7:th.24">GOBOARD7:24</a></span>;<br/><a NAME="E3:35_1_1_22_1_3_1_1_4"/>
<font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E91">A700</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E11:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E98">A707</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E104">A713</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E102">A711</font></span></a>, <a class="txt" href="#E12:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E99">A708</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E121">A730</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E26:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E112">A721</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E115">A724</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E117">A726</font></span></a>, <a class="txt" href="#E30:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E116">A725</font></span></a>, <a class="txt" href="#E19:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E106">A715</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3_1_1_4"><span class="lab"><font color="Green" title="E123">A736</font></span></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_22_1_3_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E62:35_1_1"><span class="lab"><font color="Green" title="E75">A615</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E89">A698</font></span></a>, <a class="txt" href="#E18:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E105">A714</font></span></a>, <a class="txt" href="#E4:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E92">A701</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E120">A729</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3_1_1_4"><span class="lab"><font color="Green" title="E123">A736</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_3_1_1_4"><span class="lab"><font color="Green" title="E124">A737</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_3_1_1_5"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_3_1_1_5"/><span class="lab"><font color="Green" title="E123">A738</font></span>: 
( <font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> &amp; <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_3_1_1_5"/><span class="kw">then </span>
( ( <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i1</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j19</font> ) or ( <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i1</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1</font> ) or ( <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j19</font> ) or ( <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E91">A700</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E11:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E98">A707</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E102">A711</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E100">A709</font></span></a>, <a class="txt" href="#E16:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E103">A712</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E121">A730</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E26:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E112">A721</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E115">A724</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E119">A728</font></span></a>, <a class="txt" href="#E27:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E113">A722</font></span></a>, <a class="txt" href="#E19:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E106">A715</font></span></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_22_1_3_1_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E69:35_1_1"><span class="lab"><font color="Green" title="E81">A630</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E4:35_1_1_22"><span class="lab"><font color="Green" title="E85">A636</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E8:35_1_1_22"><span class="lab"><font color="Green" title="E88">A639</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E89">A698</font></span></a>, <a class="txt" href="#E4:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E92">A701</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E120">A729</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3_1_1_5"><span class="lab"><font color="Green" title="E123">A738</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_3_1_1_6"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_3_1_1_6"/><span class="lab"><font color="Green" title="E123">A739</font></span>: 
( <font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> &amp; <font color="Maroon" title="c23">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_3_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A740</font></span>: 
( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i19</font> or <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i19</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E91">A700</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E11:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E98">A707</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E102">A711</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E100">A709</font></span></a>, <a class="txt" href="#E16:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E103">A712</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E121">A730</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E26:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E112">A721</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E119">A728</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E117">A726</font></span></a>, <a class="txt" href="#E32:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E118">A727</font></span></a>, <a class="txt" href="#E19:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E106">A715</font></span></a>, <a class="ref" href="goboard7.html#T25" title="GOBOARD7:th.25">GOBOARD7:25</a></span>;<br/><a NAME="E3:35_1_1_22_1_3_1_1_6"/>
<font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E91">A700</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E11:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E98">A707</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E102">A711</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E100">A709</font></span></a>, <a class="txt" href="#E16:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E103">A712</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E121">A730</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E26:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E112">A721</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E119">A728</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E117">A726</font></span></a>, <a class="txt" href="#E32:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E118">A727</font></span></a>, <a class="txt" href="#E19:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E106">A715</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3_1_1_6"><span class="lab"><font color="Green" title="E123">A739</font></span></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_22_1_3_1_1_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:35_1_1"><span class="lab"><font color="Green" title="E81">A630</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E4:35_1_1_22"><span class="lab"><font color="Green" title="E85">A636</font></span></a>, <a class="txt" href="#E8:35_1_1_22"><span class="lab"><font color="Green" title="E88">A639</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E89">A698</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E120">A729</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3_1_1_6"><span class="lab"><font color="Green" title="E123">A739</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_3_1_1_6"><span class="lab"><font color="Green" title="E124">A740</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_3_1_1_7"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_3_1_1_7"/><span class="lab"><font color="Green" title="E123">A741</font></span>: 
( <font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> &amp; <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_3_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A742</font></span>: 
( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> or <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E91">A700</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E11:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E98">A707</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E102">A711</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E100">A709</font></span></a>, <a class="txt" href="#E16:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E103">A712</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E121">A730</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E115">A724</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E119">A728</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E117">A726</font></span></a>, <a class="txt" href="#E28:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E114">A723</font></span></a>, <a class="txt" href="#E19:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E106">A715</font></span></a>, <a class="ref" href="goboard7.html#T25" title="GOBOARD7:th.25">GOBOARD7:25</a></span>;<br/><a NAME="E3:35_1_1_22_1_3_1_1_7"/>
<font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E91">A700</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E11:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E98">A707</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E102">A711</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E100">A709</font></span></a>, <a class="txt" href="#E16:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E103">A712</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E121">A730</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E115">A724</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E119">A728</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E117">A726</font></span></a>, <a class="txt" href="#E28:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E114">A723</font></span></a>, <a class="txt" href="#E19:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E106">A715</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3_1_1_7"><span class="lab"><font color="Green" title="E123">A741</font></span></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_22_1_3_1_1_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E69:35_1_1"><span class="lab"><font color="Green" title="E81">A630</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E8:35_1_1_22"><span class="lab"><font color="Green" title="E88">A639</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E89">A698</font></span></a>, <a class="txt" href="#E18:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E105">A714</font></span></a>, <a class="txt" href="#E4:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E92">A701</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E120">A729</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3_1_1_7"><span class="lab"><font color="Green" title="E123">A741</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_3_1_1_7"><span class="lab"><font color="Green" title="E124">A742</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_3_1_1_8"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_3_1_1_8"/><span class="lab"><font color="Green" title="E123">A743</font></span>: 
( <font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> &amp; <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_3_1_1_8"/><span class="kw">then </span>
( ( <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i1</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> ) or ( <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i1</font> &amp; <font color="Maroon" title="c26">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1</font> ) or ( <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> ) or ( <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c26">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E91">A700</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E11:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E98">A707</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E102">A711</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E100">A709</font></span></a>, <a class="txt" href="#E16:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E103">A712</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E121">A730</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E26:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E112">A721</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E115">A724</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E117">A726</font></span></a>, <a class="txt" href="#E30:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E116">A725</font></span></a>, <a class="txt" href="#E19:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E106">A715</font></span></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_22_1_3_1_1_8"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E69:35_1_1"><span class="lab"><font color="Green" title="E81">A630</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E8:35_1_1_22"><span class="lab"><font color="Green" title="E88">A639</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E89">A698</font></span></a>, <a class="txt" href="#E18:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E105">A714</font></span></a>, <a class="txt" href="#E4:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E92">A701</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E120">A729</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3_1_1_8"><span class="lab"><font color="Green" title="E123">A743</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_3_1_1_9"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_3_1_1_9"/><span class="lab"><font color="Green" title="E123">A744</font></span>: 
( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> &amp; <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_3_1_1_9"/><span class="kw">then </span>
( ( <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1</font> ) or ( <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1</font> ) or ( <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1</font> ) or ( <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E91">A700</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E104">A713</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E102">A711</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E100">A709</font></span></a>, <a class="txt" href="#E36:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E122">A731</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E121">A730</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E26:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E112">A721</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E115">A724</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E119">A728</font></span></a>, <a class="txt" href="#E27:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E113">A722</font></span></a>, <a class="txt" href="#E19:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E106">A715</font></span></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_22_1_3_1_1_9"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E69:35_1_1"><span class="lab"><font color="Green" title="E81">A630</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E8:35_1_1_22"><span class="lab"><font color="Green" title="E88">A639</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E89">A698</font></span></a>, <a class="txt" href="#E18:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E105">A714</font></span></a>, <a class="txt" href="#E4:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E92">A701</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E120">A729</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3_1_1_9"><span class="lab"><font color="Green" title="E123">A744</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_3_1_1_10"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_3_1_1_10"/><span class="lab"><font color="Green" title="E123">A745</font></span>: 
( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> &amp; <font color="Maroon" title="c23">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_3_1_1_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A746</font></span>: 
( <font color="Maroon" title="c21">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i19</font> or <font color="Maroon" title="c21">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i19</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E91">A700</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E104">A713</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E102">A711</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E100">A709</font></span></a>, <a class="txt" href="#E36:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E122">A731</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E121">A730</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E26:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E112">A721</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E119">A728</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E117">A726</font></span></a>, <a class="txt" href="#E32:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E118">A727</font></span></a>, <a class="txt" href="#E19:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E106">A715</font></span></a>, <a class="ref" href="goboard7.html#T25" title="GOBOARD7:th.25">GOBOARD7:25</a></span>;<br/><a NAME="E3:35_1_1_22_1_3_1_1_10"/>
<font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E91">A700</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E104">A713</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E102">A711</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E100">A709</font></span></a>, <a class="txt" href="#E36:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E122">A731</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E121">A730</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E26:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E112">A721</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E119">A728</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E117">A726</font></span></a>, <a class="txt" href="#E32:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E118">A727</font></span></a>, <a class="txt" href="#E19:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E106">A715</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3_1_1_10"><span class="lab"><font color="Green" title="E123">A745</font></span></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_22_1_3_1_1_10"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E62:35_1_1"><span class="lab"><font color="Green" title="E75">A615</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E89">A698</font></span></a>, <a class="txt" href="#E18:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E105">A714</font></span></a>, <a class="txt" href="#E4:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E92">A701</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E120">A729</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3_1_1_10"><span class="lab"><font color="Green" title="E123">A745</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_3_1_1_10"><span class="lab"><font color="Green" title="E124">A746</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_3_1_1_11"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_3_1_1_11"/><span class="lab"><font color="Green" title="E123">A747</font></span>: 
( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> &amp; <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_3_1_1_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A748</font></span>: 
( <font color="Maroon" title="c21">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> or <font color="Maroon" title="c21">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E91">A700</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E104">A713</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E102">A711</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E100">A709</font></span></a>, <a class="txt" href="#E36:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E122">A731</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E121">A730</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E115">A724</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E119">A728</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E117">A726</font></span></a>, <a class="txt" href="#E28:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E114">A723</font></span></a>, <a class="txt" href="#E19:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E106">A715</font></span></a>, <a class="ref" href="goboard7.html#T25" title="GOBOARD7:th.25">GOBOARD7:25</a></span>;<br/><a NAME="E3:35_1_1_22_1_3_1_1_11"/>
<font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E91">A700</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E104">A713</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E102">A711</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E100">A709</font></span></a>, <a class="txt" href="#E36:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E122">A731</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E121">A730</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E115">A724</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E119">A728</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E117">A726</font></span></a>, <a class="txt" href="#E28:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E114">A723</font></span></a>, <a class="txt" href="#E19:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E106">A715</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3_1_1_11"><span class="lab"><font color="Green" title="E123">A747</font></span></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_22_1_3_1_1_11"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:35_1_1"><span class="lab"><font color="Green" title="E81">A630</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E8:35_1_1_22"><span class="lab"><font color="Green" title="E88">A639</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E89">A698</font></span></a>, <a class="txt" href="#E18:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E105">A714</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E120">A729</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3_1_1_11"><span class="lab"><font color="Green" title="E123">A747</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_3_1_1_11"><span class="lab"><font color="Green" title="E124">A748</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_3_1_1_12"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_3_1_1_12"/><span class="lab"><font color="Green" title="E123">A749</font></span>: 
( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> &amp; <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_3_1_1_12"/><span class="kw">then </span>
( ( <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c26">j29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1</font> ) or ( <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c26">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1</font> ) or ( <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c26">j29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1</font> ) or ( <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c26">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E91">A700</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E104">A713</font></span></a>, <a class="txt" href="#E15:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E102">A711</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E100">A709</font></span></a>, <a class="txt" href="#E36:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E122">A731</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E121">A730</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E26:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E112">A721</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E115">A724</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E117">A726</font></span></a>, <a class="txt" href="#E30:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E116">A725</font></span></a>, <a class="txt" href="#E19:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E106">A715</font></span></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_22_1_3_1_1_12"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E69:35_1_1"><span class="lab"><font color="Green" title="E81">A630</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E8:35_1_1_22"><span class="lab"><font color="Green" title="E88">A639</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E89">A698</font></span></a>, <a class="txt" href="#E18:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E105">A714</font></span></a>, <a class="txt" href="#E4:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E92">A701</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E120">A729</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3_1_1_12"><span class="lab"><font color="Green" title="E123">A749</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_3_1_1_13"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_3_1_1_13"/><span class="lab"><font color="Green" title="E123">A750</font></span>: 
( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_3_1_1_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A751</font></span>: 
( <font color="Maroon" title="c22">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j19</font> or <font color="Maroon" title="c22">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j19</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E91">A700</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E11:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E98">A707</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E104">A713</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E100">A709</font></span></a>, <a class="txt" href="#E14:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E101">A710</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E121">A730</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E26:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E112">A721</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E115">A724</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E119">A728</font></span></a>, <a class="txt" href="#E27:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E113">A722</font></span></a>, <a class="txt" href="#E19:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E106">A715</font></span></a>, <a class="ref" href="goboard7.html#T24" title="GOBOARD7:th.24">GOBOARD7:24</a></span>;<br/><a NAME="E3:35_1_1_22_1_3_1_1_13"/>
<font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E91">A700</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E11:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E98">A707</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E104">A713</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E100">A709</font></span></a>, <a class="txt" href="#E14:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E101">A710</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E121">A730</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E26:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E112">A721</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E115">A724</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E119">A728</font></span></a>, <a class="txt" href="#E27:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E113">A722</font></span></a>, <a class="txt" href="#E19:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E106">A715</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3_1_1_13"><span class="lab"><font color="Green" title="E123">A750</font></span></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_22_1_3_1_1_13"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E62:35_1_1"><span class="lab"><font color="Green" title="E75">A615</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E89">A698</font></span></a>, <a class="txt" href="#E18:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E105">A714</font></span></a>, <a class="txt" href="#E4:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E92">A701</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E120">A729</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3_1_1_13"><span class="lab"><font color="Green" title="E123">A750</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_3_1_1_13"><span class="lab"><font color="Green" title="E124">A751</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_3_1_1_14"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_3_1_1_14"/><span class="lab"><font color="Green" title="E123">A752</font></span>: 
( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c23">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_3_1_1_14"/><span class="kw">then </span>
( ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i19</font> &amp; <font color="Maroon" title="c22">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j19</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i19</font> &amp; <font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j19</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j19</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j19</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E91">A700</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E11:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E98">A707</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E104">A713</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E100">A709</font></span></a>, <a class="txt" href="#E14:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E101">A710</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E121">A730</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E26:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E112">A721</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E119">A728</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E117">A726</font></span></a>, <a class="txt" href="#E32:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E118">A727</font></span></a>, <a class="txt" href="#E19:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E106">A715</font></span></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_22_1_3_1_1_14"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E69:35_1_1"><span class="lab"><font color="Green" title="E81">A630</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E8:35_1_1_22"><span class="lab"><font color="Green" title="E88">A639</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E89">A698</font></span></a>, <a class="txt" href="#E18:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E105">A714</font></span></a>, <a class="txt" href="#E4:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E92">A701</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E120">A729</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3_1_1_14"><span class="lab"><font color="Green" title="E123">A752</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_3_1_1_15"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_3_1_1_15"/><span class="lab"><font color="Green" title="E123">A753</font></span>: 
( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_3_1_1_15"/><span class="kw">then </span>
( ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c22">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j19</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j19</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j19</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j19</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E91">A700</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E11:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E98">A707</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E104">A713</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E100">A709</font></span></a>, <a class="txt" href="#E14:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E101">A710</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E121">A730</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E115">A724</font></span></a>, <a class="txt" href="#E33:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E119">A728</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E117">A726</font></span></a>, <a class="txt" href="#E28:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E114">A723</font></span></a>, <a class="txt" href="#E19:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E106">A715</font></span></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_22_1_3_1_1_15"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1_1"><span class="lab"><font color="Green" title="E72">A581</font></span></a>, <a class="txt" href="#E69:35_1_1"><span class="lab"><font color="Green" title="E81">A630</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E3:35_1_1_22"><span class="lab"><font color="Green" title="E84">A635</font></span></a>, <a class="txt" href="#E8:35_1_1_22"><span class="lab"><font color="Green" title="E88">A639</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E89">A698</font></span></a>, <a class="txt" href="#E18:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E105">A714</font></span></a>, <a class="txt" href="#E4:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E92">A701</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E120">A729</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3_1_1_15"><span class="lab"><font color="Green" title="E123">A753</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_22_1_3_1_1_16"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_22_1_3_1_1_16"/><span class="lab"><font color="Green" title="E123">A754</font></span>: 
( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c23">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i29</font> &amp; <font color="Maroon" title="c24">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_22_1_3_1_1_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A755</font></span>: 
( <font color="Maroon" title="c22">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> or <font color="Maroon" title="c22">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E91">A700</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E11:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E98">A707</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E104">A713</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E100">A709</font></span></a>, <a class="txt" href="#E14:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E101">A710</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E121">A730</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E26:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E112">A721</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E115">A724</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E117">A726</font></span></a>, <a class="txt" href="#E30:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E116">A725</font></span></a>, <a class="txt" href="#E19:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E106">A715</font></span></a>, <a class="ref" href="goboard7.html#T24" title="GOBOARD7:th.24">GOBOARD7:24</a></span>;<br/><a NAME="E3:35_1_1_22_1_3_1_1_16"/>
<font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E91">A700</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E11:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E98">A707</font></span></a>, <a class="txt" href="#E17:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E104">A713</font></span></a>, <a class="txt" href="#E13:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E100">A709</font></span></a>, <a class="txt" href="#E14:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E101">A710</font></span></a>, <a class="txt" href="#E35:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E121">A730</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E26:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E112">A721</font></span></a>, <a class="txt" href="#E29:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E115">A724</font></span></a>, <a class="txt" href="#E31:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E117">A726</font></span></a>, <a class="txt" href="#E30:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E116">A725</font></span></a>, <a class="txt" href="#E19:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E106">A715</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3_1_1_16"><span class="lab"><font color="Green" title="E123">A754</font></span></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_22_1_3_1_1_16"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:35_1_1"><span class="lab"><font color="Green" title="E81">A630</font></span></a>, <a class="txt" href="#E1:35_1_1_22"><span class="lab"><font color="Green" title="E82">A633</font></span></a>, <a class="txt" href="#E6:35_1_1_22"><span class="lab"><font color="Green" title="E87">A638</font></span></a>, <a class="txt" href="#E8:35_1_1_22"><span class="lab"><font color="Green" title="E88">A639</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E89">A698</font></span></a>, <a class="txt" href="#E18:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E105">A714</font></span></a>, <a class="txt" href="#E34:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E120">A729</font></span></a>, <a class="txt" href="#E7:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E94">A703</font></span></a>, <a class="txt" href="#E9:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E96">A705</font></span></a>, <a class="txt" href="#E22:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E108">A717</font></span></a>, <a class="txt" href="#E24:35_1_1_22_1_3"><span class="lab"><font color="Green" title="E110">A719</font></span></a>, <a class="txt" href="#E1:35_1_1_22_1_3_1_1_16"><span class="lab"><font color="Green" title="E123">A754</font></span></a>, <a class="txt" href="#E2:35_1_1_22_1_3_1_1_16"><span class="lab"><font color="Green" title="E124">A755</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E38:35_1_1_22_1_3"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">g</font> = <font color="Maroon" title="c16">g</font> as  non  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>   <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a>  <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:35_1_1"><span class="lab"><font color="Green" title="E60">A569</font></span></a>, <a class="txt" href="#E61:35_1_1"><span class="lab"><font color="Green" title="E74">A614</font></span></a>, <a class="txt" href="#E56:35_1_1"><span class="lab"><font color="Green" title="E70">A579</font></span></a>, <a class="txt" href="#E66:35_1_1"><span class="lab"><font color="Green" title="E78">A625</font></span></a>, <a class="txt" href="#E68:35_1_1"><span class="lab"><font color="Green" title="E80">A627</font></span></a>, <a class="ref" href="finseq_6.html#D1" title="FINSEQ_6:def.1">FINSEQ_6:def 1</a>, <a class="ref" href="jordan8.html#T7" title="JORDAN8:th.7">JORDAN8:7</a></span>;<br/>
<a NAME="E72:35_1_1"/><span class="lab"><font color="Green" title="E82">A756</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">f</font> holds <br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Olive" title="b1">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_23"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">f</font> implies  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c18">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1_1_23"/><span class="lab"><font color="Green" title="E83">A757</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">f</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c18">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c19">i1</font>, <font color="Maroon" title="c20">j1</font>, <font color="Maroon" title="c21">i2</font>, <font color="Maroon" title="c22">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:35_1_1_23"/><span class="lab"><font color="Green" title="E84">A758</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i1</font>,<font color="Maroon" title="c20">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_23"/><span class="lab"><font color="Green" title="E85">A759</font></span>: 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<font color="Maroon" title="c20">j1</font>
 <span class="kw">and </span><br/><a NAME="E5:35_1_1_23"/><span class="lab"><font color="Green" title="E86">A760</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i2</font>,<font color="Maroon" title="c22">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:35_1_1_23"/><span class="lab"><font color="Green" title="E87">A761</font></span>: 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i2</font>,<font color="Maroon" title="c22">j2</font>
 <span class="kw">and </span><br/><a NAME="E7:35_1_1_23"/><span class="lab"><font color="Green" title="E88">A762</font></span>: 
( ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:35_1_1"><span class="lab"><font color="Green" title="E62">A571</font></span></a>, <a class="txt" href="#E1:35_1_1_23"><span class="lab"><font color="Green" title="E83">A757</font></span></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>;<br/>
<a NAME="E8:35_1_1_23"/><span class="lab"><font color="Green" title="E89">A763</font></span>: 
<font color="Maroon" title="c19">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_23"><span class="lab"><font color="Green" title="E84">A758</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E9:35_1_1_23"/><span class="lab"><font color="Green" title="E90">A764</font></span>: 
<font color="Maroon" title="c22">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_23"><span class="lab"><font color="Green" title="E86">A760</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E10:35_1_1_23"/><span class="lab"><font color="Green" title="E91">A765</font></span>: 
<font color="Maroon" title="c20">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_23"><span class="lab"><font color="Green" title="E84">A758</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E11:35_1_1_23"/><span class="lab"><font color="Green" title="E92">A766</font></span>: 
( <font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">j1</font> &amp; <font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c22">j2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E12:35_1_1_23"/><span class="lab"><font color="Green" title="E93">A767</font></span>: 
<font color="Maroon" title="c21">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_23"><span class="lab"><font color="Green" title="E86">A760</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E13:35_1_1_23"/><span class="lab"><font color="Green" title="E94">A768</font></span>: 
( <font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c19">i1</font> &amp; <font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c21">i2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_23_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_23_1"/>
( ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_23"><span class="lab"><font color="Green" title="E88">A762</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_23_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_23_1_1"/>
( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c18">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:35_1_1_23_1_1"/><span class="kw">then </span>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c19">i1</font>,<font color="Maroon" title="c20">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:35_1_1"><span class="lab"><font color="Green" title="E62">A571</font></span></a>, <a class="txt" href="#E1:35_1_1_23"><span class="lab"><font color="Green" title="E83">A757</font></span></a>, <a class="txt" href="#E3:35_1_1_23"><span class="lab"><font color="Green" title="E84">A758</font></span></a>, <a class="txt" href="#E4:35_1_1_23"><span class="lab"><font color="Green" title="E85">A759</font></span></a>, <a class="txt" href="#E5:35_1_1_23"><span class="lab"><font color="Green" title="E86">A760</font></span></a>, <a class="txt" href="#E6:35_1_1_23"><span class="lab"><font color="Green" title="E87">A761</font></span></a>, <a class="txt" href="#E11:35_1_1_23"><span class="lab"><font color="Green" title="E92">A766</font></span></a>, <a class="ref" href="gobrd13.html#D2" title="GOBRD13:def.2">GOBRD13:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_23_1_1"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c18">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_1_23"><span class="lab"><font color="Green" title="E89">A763</font></span></a>, <a class="txt" href="#E10:35_1_1_23"><span class="lab"><font color="Green" title="E91">A765</font></span></a>, <a class="ref" href="gobrd11.html#T35" title="GOBRD11:th.35">GOBRD11:35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_23_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_23_1_2"/><span class="lab"><font color="Green" title="E95">A769</font></span>: 
( <font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c18">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:35_1_1_23_1_2"/><span class="lab"><font color="Green" title="E96">A770</font></span>: 
<font color="Maroon" title="c20">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:35_1_1_23"><span class="lab"><font color="Green" title="E91">A765</font></span></a>, <a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>;<br/><a NAME="E3:35_1_1_23_1_2"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c19">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:35_1_1"><span class="lab"><font color="Green" title="E62">A571</font></span></a>, <a class="txt" href="#E1:35_1_1_23"><span class="lab"><font color="Green" title="E83">A757</font></span></a>, <a class="txt" href="#E3:35_1_1_23"><span class="lab"><font color="Green" title="E84">A758</font></span></a>, <a class="txt" href="#E4:35_1_1_23"><span class="lab"><font color="Green" title="E85">A759</font></span></a>, <a class="txt" href="#E5:35_1_1_23"><span class="lab"><font color="Green" title="E86">A760</font></span></a>, <a class="txt" href="#E6:35_1_1_23"><span class="lab"><font color="Green" title="E87">A761</font></span></a>, <a class="txt" href="#E13:35_1_1_23"><span class="lab"><font color="Green" title="E94">A768</font></span></a>, <a class="txt" href="#E1:35_1_1_23_1_2"><span class="lab"><font color="Green" title="E95">A769</font></span></a>, <a class="ref" href="gobrd13.html#D2" title="GOBRD13:def.2">GOBRD13:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_23_1_2"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c18">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_1_23"><span class="lab"><font color="Green" title="E89">A763</font></span></a>, <a class="txt" href="#E2:35_1_1_23_1_2"><span class="lab"><font color="Green" title="E96">A770</font></span></a>, <a class="ref" href="gobrd11.html#T35" title="GOBRD11:th.35">GOBRD11:35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_23_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_23_1_3"/>
( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c18">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:35_1_1_23_1_3"/><span class="kw">then </span>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c21">i2</font>,<font color="Maroon" title="c22">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:35_1_1"><span class="lab"><font color="Green" title="E62">A571</font></span></a>, <a class="txt" href="#E1:35_1_1_23"><span class="lab"><font color="Green" title="E83">A757</font></span></a>, <a class="txt" href="#E3:35_1_1_23"><span class="lab"><font color="Green" title="E84">A758</font></span></a>, <a class="txt" href="#E4:35_1_1_23"><span class="lab"><font color="Green" title="E85">A759</font></span></a>, <a class="txt" href="#E5:35_1_1_23"><span class="lab"><font color="Green" title="E86">A760</font></span></a>, <a class="txt" href="#E6:35_1_1_23"><span class="lab"><font color="Green" title="E87">A761</font></span></a>, <a class="txt" href="#E13:35_1_1_23"><span class="lab"><font color="Green" title="E94">A768</font></span></a>, <a class="ref" href="gobrd13.html#D2" title="GOBRD13:def.2">GOBRD13:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_23_1_3"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c18">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:35_1_1_23"><span class="lab"><font color="Green" title="E93">A767</font></span></a>, <a class="txt" href="#E9:35_1_1_23"><span class="lab"><font color="Green" title="E90">A764</font></span></a>, <a class="ref" href="gobrd11.html#T35" title="GOBRD11:th.35">GOBRD11:35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_23_1_4"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_23_1_4"/><span class="lab"><font color="Green" title="E95">A771</font></span>: 
( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c18">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:35_1_1_23_1_4"/><span class="lab"><font color="Green" title="E96">A772</font></span>: 
<font color="Maroon" title="c19">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_1_23"><span class="lab"><font color="Green" title="E89">A763</font></span></a>, <a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>;<br/><a NAME="E3:35_1_1_23_1_4"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:35_1_1"><span class="lab"><font color="Green" title="E62">A571</font></span></a>, <a class="txt" href="#E1:35_1_1_23"><span class="lab"><font color="Green" title="E83">A757</font></span></a>, <a class="txt" href="#E3:35_1_1_23"><span class="lab"><font color="Green" title="E84">A758</font></span></a>, <a class="txt" href="#E4:35_1_1_23"><span class="lab"><font color="Green" title="E85">A759</font></span></a>, <a class="txt" href="#E5:35_1_1_23"><span class="lab"><font color="Green" title="E86">A760</font></span></a>, <a class="txt" href="#E6:35_1_1_23"><span class="lab"><font color="Green" title="E87">A761</font></span></a>, <a class="txt" href="#E11:35_1_1_23"><span class="lab"><font color="Green" title="E92">A766</font></span></a>, <a class="txt" href="#E1:35_1_1_23_1_4"><span class="lab"><font color="Green" title="E95">A771</font></span></a>, <a class="ref" href="gobrd13.html#D2" title="GOBRD13:def.2">GOBRD13:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_23_1_4"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c18">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c18">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_23"><span class="lab"><font color="Green" title="E90">A764</font></span></a>, <a class="txt" href="#E2:35_1_1_23_1_4"><span class="lab"><font color="Green" title="E96">A772</font></span></a>, <a class="ref" href="gobrd11.html#T35" title="GOBRD11:th.35">GOBRD11:35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_24"><span class="kw">now </span></a><div class="add"><a NAME="E1:35_1_1_24"/><span class="lab"><font color="Green" title="E83">A773</font></span>: 
for <font color="Olive" title="b1">h</font> being non  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>   <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a>  <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a>  st  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">f</font> holds <br/>for <font color="Olive" title="b2">Comp</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b2">Comp</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">h</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  holds <br/>for <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">n</font> &amp; <font color="Olive" title="b3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">f</font> &amp; <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">Comp</font> &amp; not <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">h</font> holds <br/><font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b2">Comp</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_24_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">h</font> be  non  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>   <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a>  <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">f</font> implies for <font color="Olive" title="b1">Comp</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">Comp</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">h</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  holds <br/>for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">f</font> &amp; <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Comp</font> &amp; not <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">h</font> holds <br/><font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">Comp</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1_1_24_1"/><span class="lab"><font color="Green" title="E84">A774</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Comp</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">Comp</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">h</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  holds <br/>for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">f</font> &amp; <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Comp</font> &amp; not <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">h</font> holds <br/><font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">Comp</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c19">Comp</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">Comp</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">h</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  implies for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">f</font> &amp; <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">Comp</font> &amp; not <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">h</font> holds <br/><font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c19">Comp</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:35_1_1_24_1"/><span class="lab"><font color="Green" title="E85">A775</font></span>: 
<font color="Maroon" title="c19">Comp</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">h</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">f</font> &amp; <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">Comp</font> &amp; not <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">h</font> holds <br/><font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c19">Comp</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c20">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">n</font> &amp; <font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">f</font> &amp; <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">Comp</font> &amp; not <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">h</font> implies <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c19">Comp</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:35_1_1_24_1"/><span class="lab"><font color="Green" title="E86">A776</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">n</font> &amp; <font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">f</font> )
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_24_1"/><span class="lab"><font color="Green" title="E87">A777</font></span>: 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">Comp</font>
 <span class="kw">and </span><br/><a NAME="E5:35_1_1_24_1"/><span class="lab"><font color="Green" title="E88">A778</font></span>: 
not <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c19">Comp</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c21">rc</font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c20">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">h</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E7:35_1_1_24_1"/>
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c20">n</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:35_1_1"><span class="lab"><font color="Green" title="E62">A571</font></span></a>, <a class="txt" href="#E3:35_1_1_24_1"><span class="lab"><font color="Green" title="E86">A776</font></span></a>, <a class="ref" href="jordan9.html#T10" target="_self" title="JORDAN9:th.10">Th10</a></span>;<br/>
<a NAME="E8:35_1_1_24_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">rc</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_24_1"><span class="lab"><font color="Green" title="E88">A778</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E9:35_1_1_24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A779</font></span>: 
<font color="Maroon" title="c21">rc</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c19">Comp</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_24_1"><span class="lab"><font color="Green" title="E87">A777</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E10:35_1_1_24_1"/><span class="lab"><font color="Green" title="E90">A780</font></span>: 
<font color="Maroon" title="c21">rc</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_24_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_1_1_24_1_1"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c20">n</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E3:35_1_1_24_1"><span class="lab"><font color="Green" title="E86">A776</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:35_1_1_24_1_1"/><span class="lab"><font color="Green" title="E91">A781</font></span>: 
<font color="Maroon" title="c22">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c20">n</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_24_1_1"/><span class="lab"><font color="Green" title="E92">A782</font></span>: 
<font color="Maroon" title="c22">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">p</font> = <font color="Maroon" title="c22">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_24_1_1"><span class="lab"><font color="Green" title="E91">A781</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_24_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c24">a</font> = <font color="Maroon" title="c23">p</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">rc</font> &amp; <font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_24_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_24_1_1_1_1"/>
<font color="Maroon" title="c23">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:35_1_1_24_1_1_1_1"/><span class="lab"><font color="Green" title="E93">A783</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">j</font> &amp; <font color="Maroon" title="c25">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">f</font> )
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_24_1_1_1_1"/><span class="lab"><font color="Green" title="E94">A784</font></span>: 
<font color="Maroon" title="c23">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c25">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_24_1"><span class="lab"><font color="Green" title="E84">A774</font></span></a>, <a class="ref" href="sppol_2.html#T13" title="SPPOL_2:th.13">SPPOL_2:13</a></span>;<br/><a NAME="E5:35_1_1_24_1_1_1_1"/>
<font color="Maroon" title="c23">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c25">j</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c25">j</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E3:35_1_1_24_1_1_1_1"><span class="lab"><font color="Green" title="E93">A783</font></span></a>, <a class="txt" href="#E4:35_1_1_24_1_1_1_1"><span class="lab"><font color="Green" title="E94">A784</font></span></a>, <a class="ref" href="gobrd13.html#T30" title="GOBRD13:th.30">GOBRD13:30</a></span>;<br/><a NAME="E6:35_1_1_24_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A785</font></span>: 
<font color="Maroon" title="c23">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c25">j</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E7:35_1_1_24_1_1_1_1"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c25">j</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E3:35_1_1_24_1_1_1_1"><span class="lab"><font color="Green" title="E93">A783</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:35_1_1_24_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_24_1_1"><span class="lab"><font color="Green" title="E92">A782</font></span></a>, <a class="txt" href="#E6:35_1_1_24_1_1_1_1"><span class="lab"><font color="Green" title="E95">A785</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:35_1_1_24_1_1_1"/>
<font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">rc</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_24_1_1"><span class="lab"><font color="Green" title="E91">A781</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font></span><br/><span class="kw">thus </span><a NAME="E3:35_1_1_24_1_1_1"/>
<font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_24_1_1"><span class="lab"><font color="Green" title="E92">A782</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:35_1_1_24_1_1"/>
<font color="Maroon" title="c21">rc</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:35_1_1_24_1"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c20">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:35_1_1"><span class="lab"><font color="Green" title="E62">A571</font></span></a>, <a class="txt" href="#E3:35_1_1_24_1"><span class="lab"><font color="Green" title="E86">A776</font></span></a>, <a class="ref" href="jordan9.html#T17" target="_self" title="JORDAN9:th.17">Th17</a></span>;<br/>
<a NAME="E12:35_1_1_24_1"/><span class="kw">then </span>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c20">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_24_1"><span class="lab"><font color="Green" title="E84">A774</font></span></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>;<br/>
<a NAME="E13:35_1_1_24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A786</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c20">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">h</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T43" title="SUBSET_1:th.43">SUBSET_1:43</a></span>;<br/>
<a NAME="E14:35_1_1_24_1"/><span class="lab"><font color="Green" title="E92">A787</font></span>: 
<font color="Maroon" title="c21">rc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c20">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">h</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T32" title="SUBSET_1:th.32">SUBSET_1:32</a></span>;<br/>
<a NAME="E15:35_1_1_24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A788</font></span>: 
<font color="Maroon" title="c21">rc</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">h</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E16:35_1_1_24_1"/>
<font color="Maroon" title="c21">rc</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c20">n</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E17:35_1_1_24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A789</font></span>: 
(  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c20">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c20">n</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c21">rc</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c20">n</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:35_1_1"><span class="lab"><font color="Green" title="E62">A571</font></span></a>, <a class="txt" href="#E3:35_1_1_24_1"><span class="lab"><font color="Green" title="E86">A776</font></span></a>, <a class="ref" href="jordan9.html#T13" target="_self" title="JORDAN9:th.13">Th13</a>, <a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></span>;<br/>
<a NAME="E18:35_1_1_24_1"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c20">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:35_1_1"><span class="lab"><font color="Green" title="E62">A571</font></span></a>, <a class="txt" href="#E3:35_1_1_24_1"><span class="lab"><font color="Green" title="E86">A776</font></span></a>, <a class="ref" href="jordan9.html#T12" target="_self" title="JORDAN9:th.12">Th12</a></span>;<br/>
<a NAME="E19:35_1_1_24_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">rc</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:35_1_1_24_1"><span class="lab"><font color="Green" title="E92">A787</font></span></a>, <a class="txt" href="#E13:35_1_1_24_1"><span class="lab"><font color="Green" title="E91">A786</font></span></a>, <a class="txt" href="#E17:35_1_1_24_1"><span class="lab"><font color="Green" title="E94">A789</font></span></a>, <a class="ref" href="connsp_1.html#T19" title="CONNSP_1:th.19">CONNSP_1:19</a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/>
<a NAME="E20:35_1_1_24_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">rc</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">Comp</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_24_1"><span class="lab"><font color="Green" title="E85">A775</font></span></a>, <a class="txt" href="#E9:35_1_1_24_1"><span class="lab"><font color="Green" title="E89">A779</font></span></a>, <a class="txt" href="#E15:35_1_1_24_1"><span class="lab"><font color="Green" title="E93">A788</font></span></a>, <a class="ref" href="goboard9.html#T6" title="GOBOARD9:th.6">GOBOARD9:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:35_1_1_24_1"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c19">Comp</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:35_1_1_24_1"><span class="lab"><font color="Green" title="E90">A780</font></span></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E2:35_1_1_24"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c17">g</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sppol_1.html#T49" title="SPPOL_1:th.49">SPPOL_1:49</a></span>;<br/><a NAME="E3:35_1_1_24"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></span>;<br/><a NAME="E4:35_1_1_24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A790</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></span>;<br/><a NAME="E5:35_1_1_24"/><span class="lab"><font color="Green" title="E85">A791</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c17">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan3.html#T75" title="JORDAN3:th.75">JORDAN3:75</a></span>;<br/><a NAME="E6:35_1_1_24"/><span class="lab"><font color="Green" title="E86">A792</font></span>: 
for <font color="Olive" title="b1">j</font>, <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_24_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">j</font>, <font color="Maroon" title="c19">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">j</font> &amp; <font color="Maroon" title="c18">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">k</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c18">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">j</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_1_24_2"/><span class="lab"><font color="Green" title="E87">A793</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">j</font>
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_24_2"/><span class="lab"><font color="Green" title="E88">A794</font></span>: 
<font color="Maroon" title="c18">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c18">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">j</font></span><br/>

<a NAME="E3:35_1_1_24_2"/>
<font color="Maroon" title="c18">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c19">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a>, <a class="txt" href="#E2:35_1_1_24_2"><span class="lab"><font color="Green" title="E88">A794</font></span></a></span>;<br/>
<a NAME="E4:35_1_1_24_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></span>;<br/>
<a NAME="E5:35_1_1_24_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A795</font></span>: 
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">j</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_24_2"><span class="lab"><font color="Green" title="E87">A793</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E6:35_1_1_24_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c18">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:35_1_1"><span class="lab"><font color="Green" title="E53">A490</font></span></a>, <a class="txt" href="#E2:35_1_1_24_2"><span class="lab"><font color="Green" title="E88">A794</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:35_1_1_24_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c18">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_24_2"><span class="lab"><font color="Green" title="E89">A795</font></span></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E7:35_1_1_24"/>
<font color="Maroon" title="c12">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E8:35_1_1_24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A796</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:35_1_1"><span class="lab"><font color="Green" title="E61">A570</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E9:35_1_1_24"/><span class="lab"><font color="Green" title="E88">A797</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 holds <br/> not <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c17">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_24_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_1_1_24_3"/><span class="lab"><font color="Green" title="E89">A798</font></span>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:35_1_1"><span class="lab"><font color="Green" title="E44">A231</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c18">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">n</font> &amp; <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 implies  not <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c17">g</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:35_1_1_24_3"/><span class="lab"><font color="Green" title="E90">A799</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">n</font>
 <span class="kw">and </span><br/><a NAME="E3:35_1_1_24_3"/><span class="lab"><font color="Green" title="E91">A800</font></span>: 
<font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c17">g</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c19">p</font> = <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">n</font>;<br/>
<a NAME="E4:35_1_1_24_3"/><span class="lab"><font color="Green" title="E92">A801</font></span>: 
<font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:35_1_1"><span class="lab"><font color="Green" title="E68">A577</font></span></a>, <a class="txt" href="#E3:35_1_1_24_3"><span class="lab"><font color="Green" title="E91">A800</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E5:35_1_1_24_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A802</font></span>: 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E2:35_1_1_24_3"><span class="lab"><font color="Green" title="E90">A799</font></span></a>, <a class="ref" href="jordan9.html#T8" target="_self" title="JORDAN9:th.8">Th8</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:35_1_1_24_3"/>
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c17">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E8:35_1_1_24_3"/><span class="lab"><font color="Green" title="E94">A803</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">j</font>
 <span class="kw">and </span><br/><a NAME="E9:35_1_1_24_3"/><span class="lab"><font color="Green" title="E95">A804</font></span>: 
<font color="Maroon" title="c20">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">f</font>
 <span class="kw">and </span><br/><a NAME="E10:35_1_1_24_3"/><span class="lab"><font color="Green" title="E96">A805</font></span>: 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c20">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:35_1_1"><span class="lab"><font color="Green" title="E68">A577</font></span></a>, <a class="ref" href="jordan9.html#T9" target="_self" title="JORDAN9:th.9">Th9</a></span>;<br/>
<a NAME="E11:35_1_1_24_3"/><span class="lab"><font color="Green" title="E97">A806</font></span>: 
<font color="Maroon" title="c20">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E9:35_1_1_24_3"><span class="lab"><font color="Green" title="E95">A804</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E12:35_1_1_24_3"/><span class="lab"><font color="Green" title="E98">A807</font></span>: 
<font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_24_3"><span class="lab"><font color="Green" title="E91">A800</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E13:35_1_1_24_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A808</font></span>: 
<font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_1_24_3"><span class="lab"><font color="Green" title="E94">A803</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E14:35_1_1_24_3"/><span class="lab"><font color="Green" title="E100">A809</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:35_1_1"><span class="lab"><font color="Green" title="E61">A570</font></span></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/>
<a NAME="E15:35_1_1_24_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A810</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_1_24"><span class="lab"><font color="Green" title="E87">A796</font></span></a>, <a class="txt" href="#E8:35_1_1_24_3"><span class="lab"><font color="Green" title="E94">A803</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_24_3_1"/>
( <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">j</font> or <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E9:35_1_1_24_3"><span class="lab"><font color="Green" title="E95">A804</font></span></a>, <a class="txt" href="#E10:35_1_1_24_3"><span class="lab"><font color="Green" title="E96">A805</font></span></a>, <a class="txt" href="#E15:35_1_1_24_3"><span class="lab"><font color="Green" title="E101">A810</font></span></a>, <a class="txt" href="#E1:35_1_1_24_3"><span class="lab"><font color="Green" title="E89">A798</font></span></a>, <a class="txt" href="#E5:35_1_1_24_3"><span class="lab"><font color="Green" title="E93">A802</font></span></a>, <a class="ref" href="jordan9.html#T25" target="_self" title="JORDAN9:th.25">Th25</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_24_3_1_1"/><span class="lab"><font color="Green" title="E102">A811</font></span>: 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_24_3_1_1"/><span class="lab"><font color="Green" title="E103">A812</font></span>: 
<font color="Maroon" title="c18">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c20">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a>, <a class="txt" href="#E8:35_1_1_24_3"><span class="lab"><font color="Green" title="E94">A803</font></span></a>, <a class="txt" href="#E12:35_1_1_24_3"><span class="lab"><font color="Green" title="E98">A807</font></span></a></span>;<br/><a NAME="E3:35_1_1_24_3_1_1"/>
<font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c20">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a>, <a class="txt" href="#E13:35_1_1_24_3"><span class="lab"><font color="Green" title="E99">A808</font></span></a></span>;<br/><a NAME="E4:35_1_1_24_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A813</font></span>: 
<font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c20">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_24_3"><span class="lab"><font color="Green" title="E90">A799</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c20">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E2:35_1_1_24_3"><span class="lab"><font color="Green" title="E90">A799</font></span></a>, <a class="txt" href="#E13:35_1_1_24_3"><span class="lab"><font color="Green" title="E99">A808</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E2:35_1_1_24_3"><span class="lab"><font color="Green" title="E90">A799</font></span></a>, <a class="txt" href="#E4:35_1_1_24_3"><span class="lab"><font color="Green" title="E92">A801</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c20">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E15:35_1_1_24_3"><span class="lab"><font color="Green" title="E101">A810</font></span></a>, <a class="txt" href="#E11:35_1_1_24_3"><span class="lab"><font color="Green" title="E97">A806</font></span></a>, <a class="txt" href="#E1:35_1_1_24_3_1_1"><span class="lab"><font color="Green" title="E102">A811</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c20">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c20">j</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_24_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:35_1_1"><span class="lab"><font color="Green" title="E57">A566</font></span></a>, <a class="txt" href="#E15:35_1_1_24_3"><span class="lab"><font color="Green" title="E101">A810</font></span></a>, <a class="txt" href="#E11:35_1_1_24_3"><span class="lab"><font color="Green" title="E97">A806</font></span></a>, <a class="txt" href="#E4:35_1_1_24_3_1_1"><span class="lab"><font color="Green" title="E104">A813</font></span></a>, <a class="txt" href="#E2:35_1_1_24_3_1_1"><span class="lab"><font color="Green" title="E103">A812</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_24_3_1_2"/><span class="lab"><font color="Green" title="E102">A814</font></span>: 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_24_3_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_3_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_24_3_1_2_1_1"/>
( <font color="Maroon" title="c20">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> or <font color="Maroon" title="c20">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">k</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E9:35_1_1_24_3"><span class="lab"><font color="Green" title="E95">A804</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_3_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_24_3_1_2_1_1_1"/><span class="lab"><font color="Green" title="E103">A815</font></span>: 
<font color="Maroon" title="c20">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:35_1_1_24_3_1_2_1_1_1"/><span class="lab"><font color="Green" title="E104">A816</font></span>: 
<font color="Maroon" title="c18">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a>, <a class="txt" href="#E12:35_1_1_24_3"><span class="lab"><font color="Green" title="E98">A807</font></span></a>, <a class="txt" href="#E14:35_1_1_24_3"><span class="lab"><font color="Green" title="E100">A809</font></span></a></span>;<br/><a NAME="E3:35_1_1_24_3_1_2_1_1_1"/>
<font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a>, <a class="txt" href="#E12:35_1_1_24_3"><span class="lab"><font color="Green" title="E98">A807</font></span></a>, <a class="txt" href="#E14:35_1_1_24_3"><span class="lab"><font color="Green" title="E100">A809</font></span></a></span>;<br/><a NAME="E4:35_1_1_24_3_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A817</font></span>: 
<font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_24_3"><span class="lab"><font color="Green" title="E90">A799</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E2:35_1_1_24_3"><span class="lab"><font color="Green" title="E90">A799</font></span></a>, <a class="txt" href="#E12:35_1_1_24_3"><span class="lab"><font color="Green" title="E98">A807</font></span></a>, <a class="txt" href="#E14:35_1_1_24_3"><span class="lab"><font color="Green" title="E100">A809</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E2:35_1_1_24_3"><span class="lab"><font color="Green" title="E90">A799</font></span></a>, <a class="txt" href="#E4:35_1_1_24_3"><span class="lab"><font color="Green" title="E92">A801</font></span></a>, <a class="txt" href="#E1:35_1_1_24_3_1_2"><span class="lab"><font color="Green" title="E102">A814</font></span></a>, <a class="txt" href="#E1:35_1_1_24_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E103">A815</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:35_1_1"><span class="lab"><font color="Green" title="E60">A569</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E38:35_1_1"><span class="lab"><font color="Green" title="E61">A570</font></span></a>, <a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E63">A572</font></span></a>, <a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_24_3_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:35_1_1"><span class="lab"><font color="Green" title="E57">A566</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E38:35_1_1"><span class="lab"><font color="Green" title="E61">A570</font></span></a>, <a class="txt" href="#E42:35_1_1"><span class="lab"><font color="Green" title="E64">A573</font></span></a>, <a class="txt" href="#E4:35_1_1_24_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E105">A817</font></span></a>, <a class="txt" href="#E2:35_1_1_24_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E104">A816</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_3_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_24_3_1_2_1_1_2"/><span class="lab"><font color="Green" title="E103">A818</font></span>: 
<font color="Maroon" title="c20">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c21">l</font> = <font color="Maroon" title="c20">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/><a NAME="E2:35_1_1_24_3_1_2_1_1_2"/><span class="lab"><font color="Green" title="E104">A819</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E3:35_1_1_24_3_1_2_1_1_2"/><span class="lab"><font color="Green" title="E105">A820</font></span>: 
( <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:35_1_1_24_3"><span class="lab"><font color="Green" title="E99">A808</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/><a NAME="E4:35_1_1_24_3_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A821</font></span>: 
<font color="Maroon" title="c18">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a></span>;<br/><a NAME="E5:35_1_1_24_3_1_2_1_1_2"/><span class="lab"><font color="Green" title="E107">A822</font></span>: 
<font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_24_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E105">A820</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:35_1_1_24_3_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a></span>;<br/><a NAME="E7:35_1_1_24_3_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A823</font></span>: 
<font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_24_3"><span class="lab"><font color="Green" title="E90">A799</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E2:35_1_1_24_3"><span class="lab"><font color="Green" title="E90">A799</font></span></a>, <a class="txt" href="#E5:35_1_1_24_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E107">A822</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E2:35_1_1_24_3"><span class="lab"><font color="Green" title="E90">A799</font></span></a>, <a class="txt" href="#E4:35_1_1_24_3"><span class="lab"><font color="Green" title="E92">A801</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E1:35_1_1_24_3_1_2"><span class="lab"><font color="Green" title="E102">A814</font></span></a>, <a class="txt" href="#E1:35_1_1_24_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E103">A818</font></span></a>, <a class="txt" href="#E2:35_1_1_24_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E104">A819</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E9:35_1_1_24_3_1_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:35_1_1"><span class="lab"><font color="Green" title="E57">A566</font></span></a>, <a class="txt" href="#E1:35_1_1_24_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E103">A818</font></span></a>, <a class="txt" href="#E7:35_1_1_24_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E108">A823</font></span></a>, <a class="txt" href="#E4:35_1_1_24_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E106">A821</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_1_24_3_1_2"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E10:35_1_1_24"/>
<font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_24_4"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_1_1_24_4"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E65:35_1_1"><span class="lab"><font color="Green" title="E77">A624</font></span></a>, <a class="ref" href="xreal_1.html#T147" title="XREAL_1:th.147">XREAL_1:147</a></span>;<br/>
<a NAME="E2:35_1_1_24_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A824</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="jordan9.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<a NAME="E3:35_1_1_24_4"/><span class="lab"><font color="Green" title="E90">A825</font></span>: 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">m</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_1_24"><span class="lab"><font color="Green" title="E87">A796</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E4:35_1_1_24_4"/><span class="lab"><font color="Green" title="E91">A826</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:35_1_1"><span class="lab"><font color="Green" title="E61">A570</font></span></a>, <a class="txt" href="jordan9.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">l</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">a</font> = <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">rg</font> =  <a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">p</font> = <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c22">q</font> = <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2;<br/>
<a NAME="E5:35_1_1_24_4"/><span class="lab"><font color="Green" title="E92">A827</font></span>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:35_1_1"><span class="lab"><font color="Green" title="E76">A623</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E6:35_1_1_24_4"/>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:35_1_1"><span class="lab"><font color="Green" title="E76">A623</font></span></a>, <a class="ref" href="nat_d.html#T42" title="NAT_D:th.42">NAT_D:42</a></span>;<br/>
<a NAME="E7:35_1_1_24_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A828</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/>
<a NAME="E8:35_1_1_24_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E9:35_1_1_24_4"/><span class="kw">then </span>
<font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E10:35_1_1_24_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A829</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a></span>;<br/>
<a NAME="E11:35_1_1_24_4"/><span class="lab"><font color="Green" title="E95">A830</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:35_1_1"><span class="lab"><font color="Green" title="E76">A623</font></span></a>, <a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></span>;<br/>
<a NAME="E12:35_1_1_24_4"/><span class="kw">then </span>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T42" title="NAT_D:th.42">NAT_D:42</a></span>;<br/>
<a NAME="E13:35_1_1_24_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A831</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/>
<a NAME="E14:35_1_1_24_4"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:35_1_1_24_4"><span class="lab"><font color="Green" title="E95">A830</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E15:35_1_1_24_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A832</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/>
<a NAME="E16:35_1_1_24_4"/><span class="lab"><font color="Green" title="E98">A833</font></span>: 
 <a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:35_1_1"><span class="lab"><font color="Green" title="E78">A625</font></span></a>, <a class="ref" href="jordan9.html#T7" target="_self" title="JORDAN9:th.7">Th7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">p1</font>, <font color="Maroon" title="c24">p2</font>, <font color="Maroon" title="c25">q1</font>, <font color="Maroon" title="c26">q2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E18:35_1_1_24_4"/><span class="lab"><font color="Green" title="E99">A834</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">p1</font>,<font color="Maroon" title="c24">p2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E19:35_1_1_24_4"/><span class="lab"><font color="Green" title="E100">A835</font></span>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c23">p1</font>,<font color="Maroon" title="c24">p2</font>
 <span class="kw">and </span><br/><a NAME="E20:35_1_1_24_4"/><span class="lab"><font color="Green" title="E101">A836</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">q1</font>,<font color="Maroon" title="c26">q2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E21:35_1_1_24_4"/><span class="lab"><font color="Green" title="E102">A837</font></span>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c25">q1</font>,<font color="Maroon" title="c26">q2</font>
 <span class="kw">and </span><br/><a NAME="E22:35_1_1_24_4"/><span class="lab"><font color="Green" title="E103">A838</font></span>: 
( ( <font color="Maroon" title="c23">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q1</font> &amp; <font color="Maroon" title="c24">p2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q2</font> ) or ( <font color="Maroon" title="c23">p1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q1</font> &amp; <font color="Maroon" title="c24">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q2</font> ) or ( <font color="Maroon" title="c23">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q2</font> ) or ( <font color="Maroon" title="c23">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q1</font> &amp; <font color="Maroon" title="c24">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:35_1_1_24_4"><span class="lab"><font color="Green" title="E95">A830</font></span></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>;<br/>
<a NAME="E23:35_1_1_24_4"/><span class="lab"><font color="Green" title="E104">A839</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">p1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E24:35_1_1_24_4"/><span class="lab"><font color="Green" title="E105">A840</font></span>: 
<font color="Maroon" title="c24">p2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E25:35_1_1_24_4"/><span class="lab"><font color="Green" title="E106">A841</font></span>: 
<font color="Maroon" title="c23">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E26:35_1_1_24_4"/><span class="lab"><font color="Green" title="E107">A842</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">p2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E27:35_1_1_24_4"/><span class="lab"><font color="Green" title="E108">A843</font></span>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:35_1_1"><span class="lab"><font color="Green" title="E60">A569</font></span></a>, <a class="txt" href="#E56:35_1_1"><span class="lab"><font color="Green" title="E70">A579</font></span></a>, <a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></span>;<br/>
<a NAME="E28:35_1_1_24_4"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>;<br/>
<a NAME="E29:35_1_1_24_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A844</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_24_4"><span class="lab"><font color="Green" title="E93">A828</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E110">A845</font></span>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 = 
<font color="Maroon" title="c17">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_24_4"><span class="lab"><font color="Green" title="E89">A824</font></span></a>, <a class="ref" href="finseq_5.html#T69" title="FINSEQ_5:th.69">FINSEQ_5:69</a></span>
<br/>.= 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:35_1_1_24_4"><span class="lab"><font color="Green" title="E109">A844</font></span></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></span>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:35_1_1"><span class="lab"><font color="Green" title="E61">A570</font></span></a>, <a class="ref" href="nat_d.html#T38" title="NAT_D:th.38">NAT_D:38</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_24_4"><span class="lab"><font color="Green" title="E93">A828</font></span></a>, <a class="ref" href="nat_d.html#T37" title="NAT_D:th.37">NAT_D:37</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_24_4"><span class="lab"><font color="Green" title="E92">A827</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E52:35_1_1"><span class="lab"><font color="Green" title="E69">A578</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4"><span class="lab"><font color="Green" title="E90">A825</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c11">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1

;<br/>
<a NAME="E32:35_1_1_24_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A846</font></span>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E33:35_1_1_24_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A847</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/>
<a NAME="E34:35_1_1_24_4"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E35:35_1_1_24_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A848</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_24_4"><span class="lab"><font color="Green" title="E93">A828</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E36:35_1_1_24_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A849</font></span>: 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E32:35_1_1_24_4"><span class="lab"><font color="Green" title="E111">A846</font></span></a></span>;<br/>
<a NAME="E37:35_1_1_24_4"/><span class="lab"><font color="Green" title="E115">A850</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:35_1_1"><span class="lab"><font color="Green" title="E61">A570</font></span></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/>
<a NAME="E38:35_1_1_24_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A851</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_1_24"><span class="lab"><font color="Green" title="E87">A796</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E39:35_1_1_24_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A852</font></span>: 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E63">A572</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a></span>;<br/>
<a NAME="E40:35_1_1_24_4"/>
<font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E41:35_1_1_24_4"/><span class="kw">then </span>
<font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a></span>;<br/>
<a NAME="E42:35_1_1_24_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A853</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E43:35_1_1_24_4"/>
not <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c17">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_24"><span class="lab"><font color="Green" title="E88">A797</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a></span>;<br/>
<a NAME="E44:35_1_1_24_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A854</font></span>: 
not <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sppol_2.html#T22" title="SPPOL_2:th.22">SPPOL_2:22</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_4_3"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_24_4_3"/>
( ( <font color="Maroon" title="c23">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q1</font> &amp; <font color="Maroon" title="c24">p2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q2</font> ) or ( <font color="Maroon" title="c23">p1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q1</font> &amp; <font color="Maroon" title="c24">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q2</font> ) or ( <font color="Maroon" title="c23">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q2</font> ) or ( <font color="Maroon" title="c23">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q1</font> &amp; <font color="Maroon" title="c24">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_1_1_24_4"><span class="lab"><font color="Green" title="E103">A838</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_4_3_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_24_4_3_1"/><span class="lab"><font color="Green" title="E120">A855</font></span>: 
( <font color="Maroon" title="c23">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q1</font> &amp; <font color="Maroon" title="c24">p2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c27">a1</font>, <font color="Maroon" title="c28">a2</font>, <font color="Maroon" title="c29">p91</font>, <font color="Maroon" title="c30">p92</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:35_1_1_24_4_3_1"/><span class="lab"><font color="Green" title="E121">A856</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">a1</font>,<font color="Maroon" title="c28">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_24_4_3_1"/><span class="lab"><font color="Green" title="E122">A857</font></span>: 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c27">a1</font>,<font color="Maroon" title="c28">a2</font>
 <span class="kw">and </span><br/><a NAME="E5:35_1_1_24_4_3_1"/><span class="lab"><font color="Green" title="E123">A858</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">p91</font>,<font color="Maroon" title="c30">p92</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c29">p91</font>,<font color="Maroon" title="c30">p92</font> )
 <span class="kw">and </span><br/><a NAME="E6:35_1_1_24_4_3_1"/><span class="lab"><font color="Green" title="E124">A859</font></span>: 
( ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:35_1_1"><span class="lab"><font color="Green" title="E62">A571</font></span></a>, <a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E63">A572</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>;<br/><a NAME="E7:35_1_1_24_4_3_1"/><span class="lab"><font color="Green" title="E125">A860</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">a2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E121">A856</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><span class="kw">thus </span><a NAME="E8:35_1_1_24_4_3_1"/>
<font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_24_4_3_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_4_3_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_24_4_3_1_1_1"/>
( ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E124">A859</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_4_3_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_24_4_3_1_1_1_1"/><span class="lab"><font color="Green" title="E126">A861</font></span>: 
( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:35_1_1_24_4_3_1_1_1_1"/><span class="lab"><font color="Green" title="E127">A862</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span> &amp; <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E38:35_1_1"><span class="lab"><font color="Green" title="E61">A570</font></span></a>, <a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E63">A572</font></span></a>, <a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a></span>;<br/><a NAME="E3:35_1_1_24_4_3_1_1_1_1"/><span class="lab"><font color="Green" title="E128">A863</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:35_1_1"><span class="lab"><font color="Green" title="E53">A490</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E48:35_1_1"><span class="lab"><font color="Green" title="E67">A576</font></span></a></span>;<br/><a NAME="E4:35_1_1_24_4_3_1_1_1_1"/><span class="lab"><font color="Green" title="E129">A864</font></span>: 
<font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E123">A858</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E126">A861</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E5:35_1_1_24_4_3_1_1_1_1"/><span class="lab"><font color="Green" title="E130">A865</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><span class="lab"><font color="Green" title="E131">A866</font></span>: <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E51:35_1_1"><span class="lab"><font color="Green" title="E68">A577</font></span></a>, <a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E130">A865</font></span></a></span>
;<br/><a NAME="E7:35_1_1_24_4_3_1_1_1_1"/><span class="lab"><font color="Green" title="E132">A867</font></span>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:35_1_1"><span class="lab"><font color="Green" title="E76">A623</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E8:35_1_1_24_4_3_1_1_1_1"/><span class="lab"><font color="Green" title="E133">A868</font></span>: 
<font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E123">A858</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E126">A861</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E9:35_1_1_24_4_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">A869</font></span>: 
<font color="Maroon" title="c24">p2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">a2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">p1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c24">p2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E37:35_1_1"><span class="lab"><font color="Green" title="E60">A569</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E32:35_1_1_24_4"><span class="lab"><font color="Green" title="E111">A846</font></span></a>, <a class="txt" href="#E35:35_1_1_24_4"><span class="lab"><font color="Green" title="E113">A848</font></span></a>, <a class="txt" href="#E30:35_1_1_24_4"><span class="lab"><font color="Green" title="E110">A845</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E20:35_1_1_24_4"><span class="lab"><font color="Green" title="E101">A836</font></span></a>, <a class="txt" href="#E21:35_1_1_24_4"><span class="lab"><font color="Green" title="E102">A837</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E120">A855</font></span></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></span>
<br/>.= 
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E121">A856</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E122">A857</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E129">A864</font></span></a>, <a class="txt" href="#E8:35_1_1_24_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E133">A868</font></span></a>, <a class="txt" href="#E6:35_1_1_24_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E131">A866</font></span></a>, <a class="txt" href="#E2:35_1_1_24_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E127">A862</font></span></a>, <a class="ref" href="gobrd13.html#T35" title="GOBRD13:th.35">GOBRD13:35</a></span>
;<br/><a NAME="E11:35_1_1_24_4_3_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_1_1"><span class="lab"><font color="Green" title="E49">A458</font></span></a>, <a class="txt" href="#E8:35_1_1_24"><span class="lab"><font color="Green" title="E87">A796</font></span></a>, <a class="txt" href="#E36:35_1_1_24_4"><span class="lab"><font color="Green" title="E114">A849</font></span></a></span>;<br/><a NAME="E12:35_1_1_24_4_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E135">A870</font></span>: 
( <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c28">a2</font> &amp; <font color="Maroon" title="c14">f</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4"><span class="lab"><font color="Green" title="E91">A826</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E128">A863</font></span></a>, <a class="ref" href="gobrd13.html#T45" title="GOBRD13:th.45">GOBRD13:45</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E13:35_1_1_24_4_3_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_24_4"><span class="lab"><font color="Green" title="E89">A824</font></span></a>, <a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E136">A871</font></span>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<font color="Maroon" title="c17">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_24_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E132">A867</font></span></a>, <a class="ref" href="finseq_5.html#T69" title="FINSEQ_5:th.69">FINSEQ_5:69</a></span>
<br/>.= 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_24_4"><span class="lab"><font color="Green" title="E91">A826</font></span></a>, <a class="txt" href="#E7:35_1_1_24_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E132">A867</font></span></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></span>
;<br/><span class="lab"><font color="Green" title="E137">A872</font></span>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<font color="Maroon" title="c17">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:35_1_1"><span class="lab"><font color="Green" title="E60">A569</font></span></a>, <a class="txt" href="#E61:35_1_1"><span class="lab"><font color="Green" title="E74">A614</font></span></a>, <a class="txt" href="#E56:35_1_1"><span class="lab"><font color="Green" title="E70">A579</font></span></a>, <a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></span>
;<br/><span class="kw">set </span><font color="Maroon" title="c31">rc</font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span>;<br/><a NAME="E16:35_1_1_24_4_3_1_1_1_1"/><span class="lab"><font color="Green" title="E138">A873</font></span>: 
 <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard9.html#D2" title="GOBOARD9:def.2">GOBOARD9:def 2</a></span>;<br/><a NAME="E17:35_1_1_24_4_3_1_1_1_1"/><span class="lab"><font color="Green" title="E139">A874</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">p1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:35_1_1_24_4"><span class="lab"><font color="Green" title="E104">A839</font></span></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/><a NAME="E18:35_1_1_24_4_3_1_1_1_1"/><span class="lab"><font color="Green" title="E140">A875</font></span>: 
<font color="Maroon" title="c24">p2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_1_24_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E133">A868</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E19:35_1_1_24_4_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E141">A876</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">p1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c24">p2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E121">A856</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E122">A857</font></span></a>, <a class="txt" href="#E12:35_1_1_24_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E135">A870</font></span></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></span>;<br/><a NAME="E20:35_1_1_24_4_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E142">A877</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">p1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E21:35_1_1_24_4_3_1_1_1_1"/>
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">p1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c24">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4"><span class="lab"><font color="Green" title="E91">A826</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E121">A856</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E122">A857</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E140">A875</font></span></a>, <a class="txt" href="#E12:35_1_1_24_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E135">A870</font></span></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></span>;<br/><a NAME="E22:35_1_1_24_4_3_1_1_1_1"/><span class="kw">then </span>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">p1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c28">a2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_1_1_24_4"><span class="lab"><font color="Green" title="E98">A833</font></span></a>, <a class="txt" href="#E13:35_1_1_24_4"><span class="lab"><font color="Green" title="E96">A831</font></span></a>, <a class="txt" href="#E15:35_1_1_24_4"><span class="lab"><font color="Green" title="E97">A832</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E141">A876</font></span></a>, <a class="txt" href="#E9:35_1_1_24_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E134">A869</font></span></a>, <a class="txt" href="#E17:35_1_1_24_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E139">A874</font></span></a>, <a class="txt" href="#E14:35_1_1_24_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E136">A871</font></span></a>, <a class="txt" href="#E15:35_1_1_24_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E137">A872</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>;<br/><a NAME="E23:35_1_1_24_4_3_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:35_1_1_24_4"><span class="lab"><font color="Green" title="E106">A841</font></span></a>, <a class="txt" href="#E24:35_1_1_24_4"><span class="lab"><font color="Green" title="E105">A840</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E122">A857</font></span></a>, <a class="txt" href="#E7:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E125">A860</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E129">A864</font></span></a>, <a class="txt" href="#E8:35_1_1_24_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E133">A868</font></span></a>, <a class="txt" href="#E20:35_1_1_24_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E142">A877</font></span></a>, <a class="txt" href="#E17:35_1_1_24_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E139">A874</font></span></a>, <a class="ref" href="jordan9.html#T22" target="_self" title="JORDAN9:th.22">Th22</a></span>;<br/><a NAME="E24:35_1_1_24_4_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E143">A878</font></span>: 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:35_1_1_24_4"><span class="lab"><font color="Green" title="E119">A854</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E25:35_1_1_24_4_3_1_1_1_1"/>
( <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_24"><span class="lab"><font color="Green" title="E85">A791</font></span></a>, <a class="txt" href="#E16:35_1_1_24_4"><span class="lab"><font color="Green" title="E98">A833</font></span></a>, <a class="txt" href="#E13:35_1_1_24_4"><span class="lab"><font color="Green" title="E96">A831</font></span></a>, <a class="txt" href="#E15:35_1_1_24_4"><span class="lab"><font color="Green" title="E97">A832</font></span></a>, <a class="ref" href="jordan9.html#T29" target="_self" title="JORDAN9:th.29">Th29</a>, <a class="ref" href="sppol_2.html#T22" title="SPPOL_2:th.22">SPPOL_2:22</a></span>;<br/><span class="kw">hence </span><a NAME="E26:35_1_1_24_4_3_1_1_1_1"/>
<font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E63">A572</font></span></a>, <a class="txt" href="#E1:35_1_1_24"><span class="lab"><font color="Green" title="E83">A773</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E44:35_1_1_24_4"><span class="lab"><font color="Green" title="E119">A854</font></span></a>, <a class="txt" href="#E24:35_1_1_24_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E143">A878</font></span></a>, <a class="txt" href="#E16:35_1_1_24_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E138">A873</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_4_3_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_24_4_3_1_1_1_2"/><span class="lab"><font color="Green" title="E126">A879</font></span>: 
( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:35_1_1_24_4_3_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E123">A858</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E3:35_1_1_24_4_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A880</font></span>: 
<font color="Maroon" title="c25">q1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E120">A855</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E4:35_1_1_24_4_3_1_1_1_2"/>
<font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E123">A858</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_1_1_1_2"><span class="lab"><font color="Green" title="E126">A879</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><span class="kw">then </span> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c27">a1</font>,<font color="Maroon" title="c28">a2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E37:35_1_1"><span class="lab"><font color="Green" title="E60">A569</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E32:35_1_1_24_4"><span class="lab"><font color="Green" title="E111">A846</font></span></a>, <a class="txt" href="#E35:35_1_1_24_4"><span class="lab"><font color="Green" title="E113">A848</font></span></a>, <a class="txt" href="#E30:35_1_1_24_4"><span class="lab"><font color="Green" title="E110">A845</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E20:35_1_1_24_4"><span class="lab"><font color="Green" title="E101">A836</font></span></a>, <a class="txt" href="#E21:35_1_1_24_4"><span class="lab"><font color="Green" title="E102">A837</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E120">A855</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_1_1_1_2"><span class="lab"><font color="Green" title="E127">A880</font></span></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></span>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E63">A572</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E121">A856</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E122">A857</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E123">A858</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_1_1_1_2"><span class="lab"><font color="Green" title="E126">A879</font></span></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_24_4_3_1_1_1_2"/>
<font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E63">A572</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E36:35_1_1_24_4"><span class="lab"><font color="Green" title="E114">A849</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_4_3_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_24_4_3_1_1_1_3"/>
( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:35_1_1_24_4_3_1_1_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E123">A858</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><span class="kw">then </span> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c23">p1</font>,<font color="Maroon" title="c24">p2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E63">A572</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E121">A856</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E122">A857</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E37:35_1_1"><span class="lab"><font color="Green" title="E60">A569</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E32:35_1_1_24_4"><span class="lab"><font color="Green" title="E111">A846</font></span></a>, <a class="txt" href="#E35:35_1_1_24_4"><span class="lab"><font color="Green" title="E113">A848</font></span></a>, <a class="txt" href="#E30:35_1_1_24_4"><span class="lab"><font color="Green" title="E110">A845</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E20:35_1_1_24_4"><span class="lab"><font color="Green" title="E101">A836</font></span></a>, <a class="txt" href="#E21:35_1_1_24_4"><span class="lab"><font color="Green" title="E102">A837</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E120">A855</font></span></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_24_4_3_1_1_1_3"/>
<font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E32:35_1_1_24_4"><span class="lab"><font color="Green" title="E111">A846</font></span></a>, <a class="txt" href="#E35:35_1_1_24_4"><span class="lab"><font color="Green" title="E113">A848</font></span></a>, <a class="txt" href="#E39:35_1_1_24_4"><span class="lab"><font color="Green" title="E117">A852</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_4_3_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_24_4_3_1_1_1_4"/>
( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:35_1_1_24_4_3_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A881</font></span>: 
( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E120">A855</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E123">A858</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E51:35_1_1"><span class="lab"><font color="Green" title="E68">A577</font></span></a>, <a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E21:35_1_1_24_4"><span class="lab"><font color="Green" title="E102">A837</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_1"><span class="lab"><font color="Green" title="E122">A857</font></span></a>, <a class="txt" href="#E2:35_1_1_24_4_3_1_1_1_4"><span class="lab"><font color="Green" title="E126">A881</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E33:35_1_1_24_4"><span class="lab"><font color="Green" title="E112">A847</font></span></a>, <a class="txt" href="#E35:35_1_1_24_4"><span class="lab"><font color="Green" title="E113">A848</font></span></a>, <a class="txt" href="#E30:35_1_1_24_4"><span class="lab"><font color="Green" title="E110">A845</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_24_4_3_1_1_1_4"/>
<font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:35_1_1"><span class="lab"><font color="Green" title="E57">A566</font></span></a>, <a class="txt" href="#E33:35_1_1_24_4"><span class="lab"><font color="Green" title="E112">A847</font></span></a>, <a class="txt" href="#E35:35_1_1_24_4"><span class="lab"><font color="Green" title="E113">A848</font></span></a>, <a class="txt" href="#E42:35_1_1_24_4"><span class="lab"><font color="Green" title="E118">A853</font></span></a>, <a class="txt" href="#E10:35_1_1_24_4"><span class="lab"><font color="Green" title="E94">A829</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_4_3_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_24_4_3_2"/><span class="lab"><font color="Green" title="E120">A882</font></span>: 
( <font color="Maroon" title="c23">p1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q1</font> &amp; <font color="Maroon" title="c24">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c27">a1</font>, <font color="Maroon" title="c28">a2</font>, <font color="Maroon" title="c29">p91</font>, <font color="Maroon" title="c30">p92</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:35_1_1_24_4_3_2"/><span class="lab"><font color="Green" title="E121">A883</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">a1</font>,<font color="Maroon" title="c28">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_24_4_3_2"/><span class="lab"><font color="Green" title="E122">A884</font></span>: 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c27">a1</font>,<font color="Maroon" title="c28">a2</font>
 <span class="kw">and </span><br/><a NAME="E5:35_1_1_24_4_3_2"/><span class="lab"><font color="Green" title="E123">A885</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">p91</font>,<font color="Maroon" title="c30">p92</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c29">p91</font>,<font color="Maroon" title="c30">p92</font> )
 <span class="kw">and </span><br/><a NAME="E6:35_1_1_24_4_3_2"/><span class="lab"><font color="Green" title="E124">A886</font></span>: 
( ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:35_1_1"><span class="lab"><font color="Green" title="E62">A571</font></span></a>, <a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E63">A572</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>;<br/><a NAME="E7:35_1_1_24_4_3_2"/><span class="lab"><font color="Green" title="E125">A887</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E121">A883</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><span class="kw">thus </span><a NAME="E8:35_1_1_24_4_3_2"/>
<font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_24_4_3_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_4_3_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_24_4_3_2_1_1"/>
( ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E124">A886</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_4_3_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_24_4_3_2_1_1_1"/><span class="lab"><font color="Green" title="E126">A888</font></span>: 
( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:35_1_1_24_4_3_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E123">A885</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E3:35_1_1_24_4_3_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A889</font></span>: 
<font color="Maroon" title="c26">q2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">a2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E120">A882</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E4:35_1_1_24_4_3_2_1_1_1"/><span class="lab"><font color="Green" title="E128">A890</font></span>: 
<font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E123">A885</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_2_1_1_1"><span class="lab"><font color="Green" title="E126">A888</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c27">a1</font>,<font color="Maroon" title="c28">a2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E63">A572</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E121">A883</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E122">A884</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E123">A885</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_2_1_1_1"><span class="lab"><font color="Green" title="E126">A888</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E37:35_1_1"><span class="lab"><font color="Green" title="E60">A569</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E32:35_1_1_24_4"><span class="lab"><font color="Green" title="E111">A846</font></span></a>, <a class="txt" href="#E35:35_1_1_24_4"><span class="lab"><font color="Green" title="E113">A848</font></span></a>, <a class="txt" href="#E30:35_1_1_24_4"><span class="lab"><font color="Green" title="E110">A845</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E20:35_1_1_24_4"><span class="lab"><font color="Green" title="E101">A836</font></span></a>, <a class="txt" href="#E21:35_1_1_24_4"><span class="lab"><font color="Green" title="E102">A837</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E120">A882</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_2_1_1_1"><span class="lab"><font color="Green" title="E128">A890</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_2_1_1_1"><span class="lab"><font color="Green" title="E127">A889</font></span></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_24_4_3_2_1_1_1"/>
<font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E32:35_1_1_24_4"><span class="lab"><font color="Green" title="E111">A846</font></span></a>, <a class="txt" href="#E35:35_1_1_24_4"><span class="lab"><font color="Green" title="E113">A848</font></span></a>, <a class="txt" href="#E39:35_1_1_24_4"><span class="lab"><font color="Green" title="E117">A852</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_4_3_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_24_4_3_2_1_1_2"/><span class="lab"><font color="Green" title="E126">A891</font></span>: 
( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:35_1_1_24_4_3_2_1_1_2"/><span class="lab"><font color="Green" title="E127">A892</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><span class="lab"><font color="Green" title="E128">A893</font></span>: <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E51:35_1_1"><span class="lab"><font color="Green" title="E68">A577</font></span></a>, <a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E2:35_1_1_24_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E127">A892</font></span></a></span>
;<br/><a NAME="E4:35_1_1_24_4_3_2_1_1_2"/><span class="lab"><font color="Green" title="E129">A894</font></span>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:35_1_1"><span class="lab"><font color="Green" title="E76">A623</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E5:35_1_1_24_4_3_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_24_4"><span class="lab"><font color="Green" title="E89">A824</font></span></a>, <a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A895</font></span>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<font color="Maroon" title="c17">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_24_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E129">A894</font></span></a>, <a class="ref" href="finseq_5.html#T69" title="FINSEQ_5:th.69">FINSEQ_5:69</a></span>
<br/>.= 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_24_4"><span class="lab"><font color="Green" title="E91">A826</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E129">A894</font></span></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></span>
;<br/><span class="lab"><font color="Green" title="E131">A896</font></span>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<font color="Maroon" title="c17">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:35_1_1"><span class="lab"><font color="Green" title="E60">A569</font></span></a>, <a class="txt" href="#E61:35_1_1"><span class="lab"><font color="Green" title="E74">A614</font></span></a>, <a class="txt" href="#E56:35_1_1"><span class="lab"><font color="Green" title="E70">A579</font></span></a>, <a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></span>
;<br/><a NAME="E8:35_1_1_24_4_3_2_1_1_2"/><span class="lab"><font color="Green" title="E132">A897</font></span>: 
<font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E123">A885</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E126">A891</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E9:35_1_1_24_4_3_2_1_1_2"/><span class="lab"><font color="Green" title="E133">A898</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span> &amp; <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E38:35_1_1"><span class="lab"><font color="Green" title="E61">A570</font></span></a>, <a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E63">A572</font></span></a>, <a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a></span>;<br/><a NAME="E10:35_1_1_24_4_3_2_1_1_2"/><span class="lab"><font color="Green" title="E134">A899</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:35_1_1"><span class="lab"><font color="Green" title="E53">A490</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E48:35_1_1"><span class="lab"><font color="Green" title="E67">A576</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c31">rc</font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span>;<br/><a NAME="E11:35_1_1_24_4_3_2_1_1_2"/><span class="lab"><font color="Green" title="E135">A900</font></span>: 
( <font color="Maroon" title="c27">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c23">p1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">p1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/><a NAME="E12:35_1_1_24_4_3_2_1_1_2"/><span class="lab"><font color="Green" title="E136">A901</font></span>: 
<font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E123">A885</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E126">A891</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E13:35_1_1_24_4_3_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E137">A902</font></span>: 
<font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c23">p1</font>,<font color="Maroon" title="c24">p2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E37:35_1_1"><span class="lab"><font color="Green" title="E60">A569</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E32:35_1_1_24_4"><span class="lab"><font color="Green" title="E111">A846</font></span></a>, <a class="txt" href="#E35:35_1_1_24_4"><span class="lab"><font color="Green" title="E113">A848</font></span></a>, <a class="txt" href="#E30:35_1_1_24_4"><span class="lab"><font color="Green" title="E110">A845</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E20:35_1_1_24_4"><span class="lab"><font color="Green" title="E101">A836</font></span></a>, <a class="txt" href="#E21:35_1_1_24_4"><span class="lab"><font color="Green" title="E102">A837</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E120">A882</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>
<br/>.= 
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E121">A883</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E122">A884</font></span></a>, <a class="txt" href="#E12:35_1_1_24_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E136">A901</font></span></a>, <a class="txt" href="#E8:35_1_1_24_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E132">A897</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E128">A893</font></span></a>, <a class="txt" href="#E9:35_1_1_24_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E133">A898</font></span></a>, <a class="ref" href="gobrd13.html#T37" title="GOBRD13:th.37">GOBRD13:37</a></span>
;<br/><a NAME="E15:35_1_1_24_4_3_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_1_1"><span class="lab"><font color="Green" title="E49">A458</font></span></a>, <a class="txt" href="#E8:35_1_1_24"><span class="lab"><font color="Green" title="E87">A796</font></span></a>, <a class="txt" href="#E36:35_1_1_24_4"><span class="lab"><font color="Green" title="E114">A849</font></span></a></span>;<br/><a NAME="E16:35_1_1_24_4_3_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E138">A903</font></span>: 
<font color="Maroon" title="c14">f</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4"><span class="lab"><font color="Green" title="E91">A826</font></span></a>, <a class="txt" href="#E10:35_1_1_24_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E134">A899</font></span></a>, <a class="ref" href="gobrd13.html#T45" title="GOBRD13:th.45">GOBRD13:45</a></span>;<br/><a NAME="E17:35_1_1_24_4_3_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E139">A904</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">p1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">p2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E121">A883</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E122">A884</font></span></a>, <a class="txt" href="#E12:35_1_1_24_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E136">A901</font></span></a>, <a class="txt" href="#E11:35_1_1_24_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E135">A900</font></span></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></span>;<br/><a NAME="E18:35_1_1_24_4_3_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E140">A905</font></span>: 
<font color="Maroon" title="c24">p2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E19:35_1_1_24_4_3_2_1_1_2"/>
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c23">p1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">p2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4"><span class="lab"><font color="Green" title="E91">A826</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E121">A883</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E122">A884</font></span></a>, <a class="txt" href="#E12:35_1_1_24_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E136">A901</font></span></a>, <a class="txt" href="#E11:35_1_1_24_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E135">A900</font></span></a>, <a class="txt" href="#E16:35_1_1_24_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E138">A903</font></span></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></span>;<br/><a NAME="E20:35_1_1_24_4_3_2_1_1_2"/><span class="kw">then </span>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">p1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c28">a2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_1_1_24_4"><span class="lab"><font color="Green" title="E98">A833</font></span></a>, <a class="txt" href="#E13:35_1_1_24_4"><span class="lab"><font color="Green" title="E96">A831</font></span></a>, <a class="txt" href="#E15:35_1_1_24_4"><span class="lab"><font color="Green" title="E97">A832</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E8:35_1_1_24_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E132">A897</font></span></a>, <a class="txt" href="#E17:35_1_1_24_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E139">A904</font></span></a>, <a class="txt" href="#E6:35_1_1_24_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E130">A895</font></span></a>, <a class="txt" href="#E7:35_1_1_24_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E131">A896</font></span></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></span>;<br/><a NAME="E21:35_1_1_24_4_3_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:35_1_1_24_4"><span class="lab"><font color="Green" title="E106">A841</font></span></a>, <a class="txt" href="#E26:35_1_1_24_4"><span class="lab"><font color="Green" title="E107">A842</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E122">A884</font></span></a>, <a class="txt" href="#E7:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E125">A887</font></span></a>, <a class="txt" href="#E12:35_1_1_24_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E136">A901</font></span></a>, <a class="txt" href="#E8:35_1_1_24_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E132">A897</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E140">A905</font></span></a>, <a class="txt" href="#E13:35_1_1_24_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E137">A902</font></span></a>, <a class="ref" href="jordan9.html#T22" target="_self" title="JORDAN9:th.22">Th22</a></span>;<br/><a NAME="E22:35_1_1_24_4_3_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E141">A906</font></span>: 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:35_1_1_24_4"><span class="lab"><font color="Green" title="E119">A854</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E23:35_1_1_24_4_3_2_1_1_2"/><span class="lab"><font color="Green" title="E142">A907</font></span>: 
 <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard9.html#D2" title="GOBOARD9:def.2">GOBOARD9:def 2</a></span>;<br/><a NAME="E24:35_1_1_24_4_3_2_1_1_2"/>
( <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_24"><span class="lab"><font color="Green" title="E85">A791</font></span></a>, <a class="txt" href="#E16:35_1_1_24_4"><span class="lab"><font color="Green" title="E98">A833</font></span></a>, <a class="txt" href="#E13:35_1_1_24_4"><span class="lab"><font color="Green" title="E96">A831</font></span></a>, <a class="txt" href="#E15:35_1_1_24_4"><span class="lab"><font color="Green" title="E97">A832</font></span></a>, <a class="ref" href="jordan9.html#T29" target="_self" title="JORDAN9:th.29">Th29</a>, <a class="ref" href="sppol_2.html#T22" title="SPPOL_2:th.22">SPPOL_2:22</a></span>;<br/><span class="kw">hence </span><a NAME="E25:35_1_1_24_4_3_2_1_1_2"/>
<font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E63">A572</font></span></a>, <a class="txt" href="#E1:35_1_1_24"><span class="lab"><font color="Green" title="E83">A773</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E44:35_1_1_24_4"><span class="lab"><font color="Green" title="E119">A854</font></span></a>, <a class="txt" href="#E22:35_1_1_24_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E141">A906</font></span></a>, <a class="txt" href="#E23:35_1_1_24_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E142">A907</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_4_3_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_24_4_3_2_1_1_3"/>
( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:35_1_1_24_4_3_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A908</font></span>: 
( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E120">A882</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E123">A885</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E51:35_1_1"><span class="lab"><font color="Green" title="E68">A577</font></span></a>, <a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E21:35_1_1_24_4"><span class="lab"><font color="Green" title="E102">A837</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E122">A884</font></span></a>, <a class="txt" href="#E2:35_1_1_24_4_3_2_1_1_3"><span class="lab"><font color="Green" title="E126">A908</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E33:35_1_1_24_4"><span class="lab"><font color="Green" title="E112">A847</font></span></a>, <a class="txt" href="#E35:35_1_1_24_4"><span class="lab"><font color="Green" title="E113">A848</font></span></a>, <a class="txt" href="#E30:35_1_1_24_4"><span class="lab"><font color="Green" title="E110">A845</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_24_4_3_2_1_1_3"/>
<font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:35_1_1"><span class="lab"><font color="Green" title="E57">A566</font></span></a>, <a class="txt" href="#E33:35_1_1_24_4"><span class="lab"><font color="Green" title="E112">A847</font></span></a>, <a class="txt" href="#E35:35_1_1_24_4"><span class="lab"><font color="Green" title="E113">A848</font></span></a>, <a class="txt" href="#E42:35_1_1_24_4"><span class="lab"><font color="Green" title="E118">A853</font></span></a>, <a class="txt" href="#E10:35_1_1_24_4"><span class="lab"><font color="Green" title="E94">A829</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_4_3_2_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_24_4_3_2_1_1_4"/>
( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:35_1_1_24_4_3_2_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A909</font></span>: 
( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E123">A885</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c23">p1</font>,<font color="Maroon" title="c24">p2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E37:35_1_1"><span class="lab"><font color="Green" title="E60">A569</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E32:35_1_1_24_4"><span class="lab"><font color="Green" title="E111">A846</font></span></a>, <a class="txt" href="#E35:35_1_1_24_4"><span class="lab"><font color="Green" title="E113">A848</font></span></a>, <a class="txt" href="#E30:35_1_1_24_4"><span class="lab"><font color="Green" title="E110">A845</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E20:35_1_1_24_4"><span class="lab"><font color="Green" title="E101">A836</font></span></a>, <a class="txt" href="#E21:35_1_1_24_4"><span class="lab"><font color="Green" title="E102">A837</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E120">A882</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E63">A572</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E121">A883</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_2"><span class="lab"><font color="Green" title="E122">A884</font></span></a>, <a class="txt" href="#E2:35_1_1_24_4_3_2_1_1_4"><span class="lab"><font color="Green" title="E126">A909</font></span></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_24_4_3_2_1_1_4"/>
<font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E63">A572</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E36:35_1_1_24_4"><span class="lab"><font color="Green" title="E114">A849</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_4_3_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_24_4_3_3"/><span class="lab"><font color="Green" title="E120">A910</font></span>: 
( <font color="Maroon" title="c23">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c27">a1</font>, <font color="Maroon" title="c28">a2</font>, <font color="Maroon" title="c29">p91</font>, <font color="Maroon" title="c30">p92</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:35_1_1_24_4_3_3"/><span class="lab"><font color="Green" title="E121">A911</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">a1</font>,<font color="Maroon" title="c28">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_24_4_3_3"/><span class="lab"><font color="Green" title="E122">A912</font></span>: 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c27">a1</font>,<font color="Maroon" title="c28">a2</font>
 <span class="kw">and </span><br/><a NAME="E5:35_1_1_24_4_3_3"/><span class="lab"><font color="Green" title="E123">A913</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">p91</font>,<font color="Maroon" title="c30">p92</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c29">p91</font>,<font color="Maroon" title="c30">p92</font> )
 <span class="kw">and </span><br/><a NAME="E6:35_1_1_24_4_3_3"/><span class="lab"><font color="Green" title="E124">A914</font></span>: 
( ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:35_1_1"><span class="lab"><font color="Green" title="E62">A571</font></span></a>, <a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E63">A572</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>;<br/><a NAME="E7:35_1_1_24_4_3_3"/><span class="lab"><font color="Green" title="E125">A915</font></span>: 
<font color="Maroon" title="c27">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E121">A911</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><span class="kw">thus </span><a NAME="E8:35_1_1_24_4_3_3"/>
<font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_24_4_3_3_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_4_3_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_24_4_3_3_1_1"/>
( ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E124">A914</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_4_3_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_24_4_3_3_1_1_1"/><span class="lab"><font color="Green" title="E126">A916</font></span>: 
( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:35_1_1_24_4_3_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E123">A913</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E3:35_1_1_24_4_3_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A917</font></span>: 
<font color="Maroon" title="c26">q2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">a2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E120">A910</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E4:35_1_1_24_4_3_3_1_1_1"/>
<font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E123">A913</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_3_1_1_1"><span class="lab"><font color="Green" title="E126">A916</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E5:35_1_1_24_4_3_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A918</font></span>: 
<font color="Maroon" title="c25">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">a1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E120">A910</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c25">q1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c26">q2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E37:35_1_1"><span class="lab"><font color="Green" title="E60">A569</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E32:35_1_1_24_4"><span class="lab"><font color="Green" title="E111">A846</font></span></a>, <a class="txt" href="#E35:35_1_1_24_4"><span class="lab"><font color="Green" title="E113">A848</font></span></a>, <a class="txt" href="#E30:35_1_1_24_4"><span class="lab"><font color="Green" title="E110">A845</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E20:35_1_1_24_4"><span class="lab"><font color="Green" title="E101">A836</font></span></a>, <a class="txt" href="#E21:35_1_1_24_4"><span class="lab"><font color="Green" title="E102">A837</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E120">A910</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E63">A572</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E121">A911</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E122">A912</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E123">A913</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_3_1_1_1"><span class="lab"><font color="Green" title="E126">A916</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_3_1_1_1"><span class="lab"><font color="Green" title="E128">A918</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_3_1_1_1"><span class="lab"><font color="Green" title="E127">A917</font></span></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:35_1_1_24_4_3_3_1_1_1"/>
<font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E63">A572</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E36:35_1_1_24_4"><span class="lab"><font color="Green" title="E114">A849</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_4_3_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_24_4_3_3_1_1_2"/>
( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:35_1_1_24_4_3_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A919</font></span>: 
( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E123">A913</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E51:35_1_1"><span class="lab"><font color="Green" title="E68">A577</font></span></a>, <a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E21:35_1_1_24_4"><span class="lab"><font color="Green" title="E102">A837</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E120">A910</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E122">A912</font></span></a>, <a class="txt" href="#E2:35_1_1_24_4_3_3_1_1_2"><span class="lab"><font color="Green" title="E126">A919</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E33:35_1_1_24_4"><span class="lab"><font color="Green" title="E112">A847</font></span></a>, <a class="txt" href="#E35:35_1_1_24_4"><span class="lab"><font color="Green" title="E113">A848</font></span></a>, <a class="txt" href="#E30:35_1_1_24_4"><span class="lab"><font color="Green" title="E110">A845</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_24_4_3_3_1_1_2"/>
<font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:35_1_1"><span class="lab"><font color="Green" title="E57">A566</font></span></a>, <a class="txt" href="#E33:35_1_1_24_4"><span class="lab"><font color="Green" title="E112">A847</font></span></a>, <a class="txt" href="#E35:35_1_1_24_4"><span class="lab"><font color="Green" title="E113">A848</font></span></a>, <a class="txt" href="#E42:35_1_1_24_4"><span class="lab"><font color="Green" title="E118">A853</font></span></a>, <a class="txt" href="#E10:35_1_1_24_4"><span class="lab"><font color="Green" title="E94">A829</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_4_3_3_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_24_4_3_3_1_1_3"/><span class="lab"><font color="Green" title="E126">A920</font></span>: 
( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span><br/><div class="add"><span class="lab"><font color="Green" title="E127">A921</font></span>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<font color="Maroon" title="c17">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:35_1_1"><span class="lab"><font color="Green" title="E60">A569</font></span></a>, <a class="txt" href="#E61:35_1_1"><span class="lab"><font color="Green" title="E74">A614</font></span></a>, <a class="txt" href="#E56:35_1_1"><span class="lab"><font color="Green" title="E70">A579</font></span></a>, <a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></span>
;<br/><a NAME="E3:35_1_1_24_4_3_3_1_1_3"/><span class="lab"><font color="Green" title="E128">A922</font></span>: 
<font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E123">A913</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E126">A920</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E4:35_1_1_24_4_3_3_1_1_3"/>
<font color="Maroon" title="c23">p1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c23">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/><a NAME="E5:35_1_1_24_4_3_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A923</font></span>: 
<font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c23">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E128">A922</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:35_1_1_24_4_3_3_1_1_3"/><span class="lab"><font color="Green" title="E130">A924</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><span class="lab"><font color="Green" title="E131">A925</font></span>: <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E51:35_1_1"><span class="lab"><font color="Green" title="E68">A577</font></span></a>, <a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E6:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E130">A924</font></span></a></span>
;<br/><a NAME="E8:35_1_1_24_4_3_3_1_1_3"/><span class="lab"><font color="Green" title="E132">A926</font></span>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:35_1_1"><span class="lab"><font color="Green" title="E76">A623</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E9:35_1_1_24_4_3_3_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_24_4"><span class="lab"><font color="Green" title="E89">A824</font></span></a>, <a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A927</font></span>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<font color="Maroon" title="c17">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E132">A926</font></span></a>, <a class="ref" href="finseq_5.html#T69" title="FINSEQ_5:th.69">FINSEQ_5:69</a></span>
<br/>.= 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_24_4"><span class="lab"><font color="Green" title="E91">A826</font></span></a>, <a class="txt" href="#E8:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E132">A926</font></span></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></span>
;<br/><a NAME="E11:35_1_1_24_4_3_3_1_1_3"/><span class="lab"><font color="Green" title="E134">A928</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:35_1_1"><span class="lab"><font color="Green" title="E53">A490</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E48:35_1_1"><span class="lab"><font color="Green" title="E67">A576</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c31">rc</font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span>;<br/><a NAME="E12:35_1_1_24_4_3_3_1_1_3"/><span class="lab"><font color="Green" title="E135">A929</font></span>: 
 <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard9.html#D2" title="GOBOARD9:def.2">GOBOARD9:def 2</a></span>;<br/><a NAME="E13:35_1_1_24_4_3_3_1_1_3"/><span class="lab"><font color="Green" title="E136">A930</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c24">p2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:35_1_1_24_4"><span class="lab"><font color="Green" title="E107">A842</font></span></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/><a NAME="E14:35_1_1_24_4_3_3_1_1_3"/><span class="lab"><font color="Green" title="E137">A931</font></span>: 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E38:35_1_1"><span class="lab"><font color="Green" title="E61">A570</font></span></a>, <a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E63">A572</font></span></a>, <a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a></span>;<br/><a NAME="E15:35_1_1_24_4_3_3_1_1_3"/><span class="lab"><font color="Green" title="E138">A932</font></span>: 
( <font color="Maroon" title="c23">p1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E120">A910</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E16:35_1_1_24_4_3_3_1_1_3"/><span class="lab"><font color="Green" title="E139">A933</font></span>: 
<font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E123">A913</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E126">A920</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c25">q1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c26">q2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E37:35_1_1"><span class="lab"><font color="Green" title="E60">A569</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E32:35_1_1_24_4"><span class="lab"><font color="Green" title="E111">A846</font></span></a>, <a class="txt" href="#E35:35_1_1_24_4"><span class="lab"><font color="Green" title="E113">A848</font></span></a>, <a class="txt" href="#E30:35_1_1_24_4"><span class="lab"><font color="Green" title="E110">A845</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E20:35_1_1_24_4"><span class="lab"><font color="Green" title="E101">A836</font></span></a>, <a class="txt" href="#E21:35_1_1_24_4"><span class="lab"><font color="Green" title="E102">A837</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E120">A910</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>
<br/>.= 
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E120">A910</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E121">A911</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E122">A912</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E128">A922</font></span></a>, <a class="txt" href="#E16:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E139">A933</font></span></a>, <a class="txt" href="#E15:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E138">A932</font></span></a>, <a class="txt" href="#E7:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E131">A925</font></span></a>, <a class="txt" href="#E14:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E137">A931</font></span></a>, <a class="ref" href="gobrd13.html#T39" title="GOBRD13:th.39">GOBRD13:39</a></span>
;<br/><a NAME="E18:35_1_1_24_4_3_3_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_1_1"><span class="lab"><font color="Green" title="E49">A458</font></span></a>, <a class="txt" href="#E8:35_1_1_24"><span class="lab"><font color="Green" title="E87">A796</font></span></a>, <a class="txt" href="#E36:35_1_1_24_4"><span class="lab"><font color="Green" title="E114">A849</font></span></a></span>;<br/><a NAME="E19:35_1_1_24_4_3_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E140">A934</font></span>: 
<font color="Maroon" title="c14">f</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4"><span class="lab"><font color="Green" title="E91">A826</font></span></a>, <a class="txt" href="#E11:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E134">A928</font></span></a>, <a class="ref" href="gobrd13.html#T45" title="GOBRD13:th.45">GOBRD13:45</a></span>;<br/><a NAME="E20:35_1_1_24_4_3_3_1_1_3"/><span class="lab"><font color="Green" title="E141">A935</font></span>: 
<font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E139">A933</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E21:35_1_1_24_4_3_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E142">A936</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">p1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">p2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E121">A911</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E122">A912</font></span></a>, <a class="txt" href="#E16:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E139">A933</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E129">A923</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E140">A934</font></span></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></span>;<br/><a NAME="E22:35_1_1_24_4_3_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E143">A937</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">p2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E23:35_1_1_24_4_3_3_1_1_3"/>
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c23">p1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">p2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4"><span class="lab"><font color="Green" title="E91">A826</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E121">A911</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E122">A912</font></span></a>, <a class="txt" href="#E16:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E139">A933</font></span></a>, <a class="txt" href="#E20:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E141">A935</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E129">A923</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E140">A934</font></span></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></span>;<br/><a NAME="E24:35_1_1_24_4_3_3_1_1_3"/><span class="kw">then </span>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c23">p1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">p2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_1_1_24_4"><span class="lab"><font color="Green" title="E98">A833</font></span></a>, <a class="txt" href="#E13:35_1_1_24_4"><span class="lab"><font color="Green" title="E96">A831</font></span></a>, <a class="txt" href="#E15:35_1_1_24_4"><span class="lab"><font color="Green" title="E97">A832</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E21:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E142">A936</font></span></a>, <a class="txt" href="#E13:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E136">A930</font></span></a>, <a class="txt" href="#E10:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E133">A927</font></span></a>, <a class="txt" href="#E2:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E127">A921</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>;<br/><a NAME="E25:35_1_1_24_4_3_3_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:35_1_1_24_4"><span class="lab"><font color="Green" title="E104">A839</font></span></a>, <a class="txt" href="#E24:35_1_1_24_4"><span class="lab"><font color="Green" title="E105">A840</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E122">A912</font></span></a>, <a class="txt" href="#E7:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E125">A915</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E128">A922</font></span></a>, <a class="txt" href="#E16:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E139">A933</font></span></a>, <a class="txt" href="#E22:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E143">A937</font></span></a>, <a class="txt" href="#E13:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E136">A930</font></span></a>, <a class="ref" href="jordan9.html#T22" target="_self" title="JORDAN9:th.22">Th22</a></span>;<br/><a NAME="E26:35_1_1_24_4_3_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E144">A938</font></span>: 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:35_1_1_24_4"><span class="lab"><font color="Green" title="E119">A854</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E27:35_1_1_24_4_3_3_1_1_3"/>
( <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_24"><span class="lab"><font color="Green" title="E85">A791</font></span></a>, <a class="txt" href="#E16:35_1_1_24_4"><span class="lab"><font color="Green" title="E98">A833</font></span></a>, <a class="txt" href="#E13:35_1_1_24_4"><span class="lab"><font color="Green" title="E96">A831</font></span></a>, <a class="txt" href="#E15:35_1_1_24_4"><span class="lab"><font color="Green" title="E97">A832</font></span></a>, <a class="ref" href="jordan9.html#T29" target="_self" title="JORDAN9:th.29">Th29</a>, <a class="ref" href="sppol_2.html#T22" title="SPPOL_2:th.22">SPPOL_2:22</a></span>;<br/><span class="kw">hence </span><a NAME="E28:35_1_1_24_4_3_3_1_1_3"/>
<font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E63">A572</font></span></a>, <a class="txt" href="#E1:35_1_1_24"><span class="lab"><font color="Green" title="E83">A773</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E44:35_1_1_24_4"><span class="lab"><font color="Green" title="E119">A854</font></span></a>, <a class="txt" href="#E26:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E144">A938</font></span></a>, <a class="txt" href="#E12:35_1_1_24_4_3_3_1_1_3"><span class="lab"><font color="Green" title="E135">A929</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_4_3_3_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_24_4_3_3_1_1_4"/><span class="lab"><font color="Green" title="E126">A939</font></span>: 
( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:35_1_1_24_4_3_3_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E123">A913</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E3:35_1_1_24_4_3_3_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A940</font></span>: 
<font color="Maroon" title="c25">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">a1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E120">A910</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E4:35_1_1_24_4_3_3_1_1_4"/>
<font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E123">A913</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_3_1_1_4"><span class="lab"><font color="Green" title="E126">A939</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><span class="kw">then </span> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c25">q1</font>,<font color="Maroon" title="c26">q2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E63">A572</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E120">A910</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E121">A911</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E122">A912</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E123">A913</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_3_1_1_4"><span class="lab"><font color="Green" title="E126">A939</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_3_1_1_4"><span class="lab"><font color="Green" title="E127">A940</font></span></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></span>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E37:35_1_1"><span class="lab"><font color="Green" title="E60">A569</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E32:35_1_1_24_4"><span class="lab"><font color="Green" title="E111">A846</font></span></a>, <a class="txt" href="#E35:35_1_1_24_4"><span class="lab"><font color="Green" title="E113">A848</font></span></a>, <a class="txt" href="#E30:35_1_1_24_4"><span class="lab"><font color="Green" title="E110">A845</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E20:35_1_1_24_4"><span class="lab"><font color="Green" title="E101">A836</font></span></a>, <a class="txt" href="#E21:35_1_1_24_4"><span class="lab"><font color="Green" title="E102">A837</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_3"><span class="lab"><font color="Green" title="E120">A910</font></span></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_24_4_3_3_1_1_4"/>
<font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E32:35_1_1_24_4"><span class="lab"><font color="Green" title="E111">A846</font></span></a>, <a class="txt" href="#E35:35_1_1_24_4"><span class="lab"><font color="Green" title="E113">A848</font></span></a>, <a class="txt" href="#E39:35_1_1_24_4"><span class="lab"><font color="Green" title="E117">A852</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_4_3_4"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_24_4_3_4"/><span class="lab"><font color="Green" title="E120">A941</font></span>: 
( <font color="Maroon" title="c23">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q1</font> &amp; <font color="Maroon" title="c24">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c27">a1</font>, <font color="Maroon" title="c28">a2</font>, <font color="Maroon" title="c29">p91</font>, <font color="Maroon" title="c30">p92</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:35_1_1_24_4_3_4"/><span class="lab"><font color="Green" title="E121">A942</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">a1</font>,<font color="Maroon" title="c28">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_24_4_3_4"/><span class="lab"><font color="Green" title="E122">A943</font></span>: 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c27">a1</font>,<font color="Maroon" title="c28">a2</font>
 <span class="kw">and </span><br/><a NAME="E5:35_1_1_24_4_3_4"/><span class="lab"><font color="Green" title="E123">A944</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">p91</font>,<font color="Maroon" title="c30">p92</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c29">p91</font>,<font color="Maroon" title="c30">p92</font> )
 <span class="kw">and </span><br/><a NAME="E6:35_1_1_24_4_3_4"/><span class="lab"><font color="Green" title="E124">A945</font></span>: 
( ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:35_1_1"><span class="lab"><font color="Green" title="E62">A571</font></span></a>, <a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E63">A572</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>;<br/><a NAME="E7:35_1_1_24_4_3_4"/><span class="lab"><font color="Green" title="E125">A946</font></span>: 
<font color="Maroon" title="c28">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E121">A942</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><span class="kw">thus </span><a NAME="E8:35_1_1_24_4_3_4"/>
<font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_24_4_3_4_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_4_3_4_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_1_24_4_3_4_1_1"/>
( ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E124">A945</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_4_3_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_24_4_3_4_1_1_1"/>
( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:35_1_1_24_4_3_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A947</font></span>: 
( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E123">A944</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E51:35_1_1"><span class="lab"><font color="Green" title="E68">A577</font></span></a>, <a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E21:35_1_1_24_4"><span class="lab"><font color="Green" title="E102">A837</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E120">A941</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E122">A943</font></span></a>, <a class="txt" href="#E2:35_1_1_24_4_3_4_1_1_1"><span class="lab"><font color="Green" title="E126">A947</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E33:35_1_1_24_4"><span class="lab"><font color="Green" title="E112">A847</font></span></a>, <a class="txt" href="#E35:35_1_1_24_4"><span class="lab"><font color="Green" title="E113">A848</font></span></a>, <a class="txt" href="#E30:35_1_1_24_4"><span class="lab"><font color="Green" title="E110">A845</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_24_4_3_4_1_1_1"/>
<font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:35_1_1"><span class="lab"><font color="Green" title="E57">A566</font></span></a>, <a class="txt" href="#E33:35_1_1_24_4"><span class="lab"><font color="Green" title="E112">A847</font></span></a>, <a class="txt" href="#E35:35_1_1_24_4"><span class="lab"><font color="Green" title="E113">A848</font></span></a>, <a class="txt" href="#E42:35_1_1_24_4"><span class="lab"><font color="Green" title="E118">A853</font></span></a>, <a class="txt" href="#E10:35_1_1_24_4"><span class="lab"><font color="Green" title="E94">A829</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_4_3_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_24_4_3_4_1_1_2"/><span class="lab"><font color="Green" title="E126">A948</font></span>: 
( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:35_1_1_24_4_3_4_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E123">A944</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E3:35_1_1_24_4_3_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A949</font></span>: 
<font color="Maroon" title="c28">a2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E120">A941</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E4:35_1_1_24_4_3_4_1_1_2"/>
<font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E123">A944</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_4_1_1_2"><span class="lab"><font color="Green" title="E126">A948</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E5:35_1_1_24_4_3_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A950</font></span>: 
<font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E120">A941</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c27">a1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c28">a2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E63">A572</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E121">A942</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E122">A943</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E123">A944</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_4_1_1_2"><span class="lab"><font color="Green" title="E126">A948</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E37:35_1_1"><span class="lab"><font color="Green" title="E60">A569</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E32:35_1_1_24_4"><span class="lab"><font color="Green" title="E111">A846</font></span></a>, <a class="txt" href="#E35:35_1_1_24_4"><span class="lab"><font color="Green" title="E113">A848</font></span></a>, <a class="txt" href="#E30:35_1_1_24_4"><span class="lab"><font color="Green" title="E110">A845</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E20:35_1_1_24_4"><span class="lab"><font color="Green" title="E101">A836</font></span></a>, <a class="txt" href="#E21:35_1_1_24_4"><span class="lab"><font color="Green" title="E102">A837</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E120">A941</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_4_1_1_2"><span class="lab"><font color="Green" title="E128">A950</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_4_1_1_2"><span class="lab"><font color="Green" title="E127">A949</font></span></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:35_1_1_24_4_3_4_1_1_2"/>
<font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E32:35_1_1_24_4"><span class="lab"><font color="Green" title="E111">A846</font></span></a>, <a class="txt" href="#E35:35_1_1_24_4"><span class="lab"><font color="Green" title="E113">A848</font></span></a>, <a class="txt" href="#E39:35_1_1_24_4"><span class="lab"><font color="Green" title="E117">A852</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_4_3_4_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_24_4_3_4_1_1_3"/><span class="lab"><font color="Green" title="E126">A951</font></span>: 
( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:35_1_1_24_4_3_4_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E123">A944</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E3:35_1_1_24_4_3_4_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A952</font></span>: 
<font color="Maroon" title="c28">a2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E120">A941</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E4:35_1_1_24_4_3_4_1_1_3"/><span class="lab"><font color="Green" title="E128">A953</font></span>: 
<font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E123">A944</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_4_1_1_3"><span class="lab"><font color="Green" title="E126">A951</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c25">q1</font>,<font color="Maroon" title="c26">q2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E37:35_1_1"><span class="lab"><font color="Green" title="E60">A569</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E32:35_1_1_24_4"><span class="lab"><font color="Green" title="E111">A846</font></span></a>, <a class="txt" href="#E35:35_1_1_24_4"><span class="lab"><font color="Green" title="E113">A848</font></span></a>, <a class="txt" href="#E30:35_1_1_24_4"><span class="lab"><font color="Green" title="E110">A845</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E20:35_1_1_24_4"><span class="lab"><font color="Green" title="E101">A836</font></span></a>, <a class="txt" href="#E21:35_1_1_24_4"><span class="lab"><font color="Green" title="E102">A837</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E120">A941</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E63">A572</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E120">A941</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E121">A942</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E122">A943</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E123">A944</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_4_1_1_3"><span class="lab"><font color="Green" title="E126">A951</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_4_1_1_3"><span class="lab"><font color="Green" title="E128">A953</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_4_1_1_3"><span class="lab"><font color="Green" title="E127">A952</font></span></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_24_4_3_4_1_1_3"/>
<font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E63">A572</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E36:35_1_1_24_4"><span class="lab"><font color="Green" title="E114">A849</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_24_4_3_4_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_24_4_3_4_1_1_4"/><span class="lab"><font color="Green" title="E126">A954</font></span>: 
( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p91</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p92</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c31">rc</font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span>;<br/><a NAME="E2:35_1_1_24_4_3_4_1_1_4"/><span class="lab"><font color="Green" title="E127">A955</font></span>: 
 <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard9.html#D2" title="GOBOARD9:def.2">GOBOARD9:def 2</a></span>;<br/><a NAME="E3:35_1_1_24_4_3_4_1_1_4"/><span class="lab"><font color="Green" title="E128">A956</font></span>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:35_1_1"><span class="lab"><font color="Green" title="E76">A623</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E4:35_1_1_24_4_3_4_1_1_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_24_4"><span class="lab"><font color="Green" title="E89">A824</font></span></a>, <a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A957</font></span>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<font color="Maroon" title="c17">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_24_4_3_4_1_1_4"><span class="lab"><font color="Green" title="E128">A956</font></span></a>, <a class="ref" href="finseq_5.html#T69" title="FINSEQ_5:th.69">FINSEQ_5:69</a></span>
<br/>.= 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_24_4"><span class="lab"><font color="Green" title="E91">A826</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_4_1_1_4"><span class="lab"><font color="Green" title="E128">A956</font></span></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></span>
;<br/><span class="lab"><font color="Green" title="E130">A958</font></span>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<font color="Maroon" title="c17">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:35_1_1"><span class="lab"><font color="Green" title="E60">A569</font></span></a>, <a class="txt" href="#E61:35_1_1"><span class="lab"><font color="Green" title="E74">A614</font></span></a>, <a class="txt" href="#E56:35_1_1"><span class="lab"><font color="Green" title="E70">A579</font></span></a>, <a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></span>
;<br/><a NAME="E7:35_1_1_24_4_3_4_1_1_4"/><span class="lab"><font color="Green" title="E131">A959</font></span>: 
<font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E123">A944</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_4_1_1_4"><span class="lab"><font color="Green" title="E126">A954</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E8:35_1_1_24_4_3_4_1_1_4"/><span class="lab"><font color="Green" title="E132">A960</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><span class="lab"><font color="Green" title="E133">A961</font></span>: <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E51:35_1_1"><span class="lab"><font color="Green" title="E68">A577</font></span></a>, <a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E8:35_1_1_24_4_3_4_1_1_4"><span class="lab"><font color="Green" title="E132">A960</font></span></a></span>
;<br/><a NAME="E10:35_1_1_24_4_3_4_1_1_4"/><span class="lab"><font color="Green" title="E134">A962</font></span>: 
( <font color="Maroon" title="c24">p2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E42">A192</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E120">A941</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E11:35_1_1_24_4_3_4_1_1_4"/><span class="lab"><font color="Green" title="E135">A963</font></span>: 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E38:35_1_1"><span class="lab"><font color="Green" title="E61">A570</font></span></a>, <a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E63">A572</font></span></a>, <a class="txt" href="#E6:35_1_1_24"><span class="lab"><font color="Green" title="E86">A792</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a></span>;<br/><a NAME="E12:35_1_1_24_4_3_4_1_1_4"/><span class="lab"><font color="Green" title="E136">A964</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:35_1_1"><span class="lab"><font color="Green" title="E53">A490</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E48:35_1_1"><span class="lab"><font color="Green" title="E67">A576</font></span></a></span>;<br/><a NAME="E13:35_1_1_24_4_3_4_1_1_4"/><span class="lab"><font color="Green" title="E137">A965</font></span>: 
<font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E123">A944</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_4_1_1_4"><span class="lab"><font color="Green" title="E126">A954</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c25">q1</font>,<font color="Maroon" title="c26">q2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E37:35_1_1"><span class="lab"><font color="Green" title="E60">A569</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E32:35_1_1_24_4"><span class="lab"><font color="Green" title="E111">A846</font></span></a>, <a class="txt" href="#E35:35_1_1_24_4"><span class="lab"><font color="Green" title="E113">A848</font></span></a>, <a class="txt" href="#E30:35_1_1_24_4"><span class="lab"><font color="Green" title="E110">A845</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E20:35_1_1_24_4"><span class="lab"><font color="Green" title="E101">A836</font></span></a>, <a class="txt" href="#E21:35_1_1_24_4"><span class="lab"><font color="Green" title="E102">A837</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E120">A941</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>
<br/>.= 
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E1:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E120">A941</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E121">A942</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E122">A943</font></span></a>, <a class="txt" href="#E7:35_1_1_24_4_3_4_1_1_4"><span class="lab"><font color="Green" title="E131">A959</font></span></a>, <a class="txt" href="#E13:35_1_1_24_4_3_4_1_1_4"><span class="lab"><font color="Green" title="E137">A965</font></span></a>, <a class="txt" href="#E10:35_1_1_24_4_3_4_1_1_4"><span class="lab"><font color="Green" title="E134">A962</font></span></a>, <a class="txt" href="#E9:35_1_1_24_4_3_4_1_1_4"><span class="lab"><font color="Green" title="E133">A961</font></span></a>, <a class="txt" href="#E11:35_1_1_24_4_3_4_1_1_4"><span class="lab"><font color="Green" title="E135">A963</font></span></a>, <a class="ref" href="gobrd13.html#T41" title="GOBRD13:th.41">GOBRD13:41</a></span>
;<br/><a NAME="E15:35_1_1_24_4_3_4_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_1_1"><span class="lab"><font color="Green" title="E49">A458</font></span></a>, <a class="txt" href="#E8:35_1_1_24"><span class="lab"><font color="Green" title="E87">A796</font></span></a>, <a class="txt" href="#E36:35_1_1_24_4"><span class="lab"><font color="Green" title="E114">A849</font></span></a></span>;<br/><a NAME="E16:35_1_1_24_4_3_4_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E138">A966</font></span>: 
( <font color="Maroon" title="c24">p2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">p2</font> &amp; <font color="Maroon" title="c14">f</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4"><span class="lab"><font color="Green" title="E91">A826</font></span></a>, <a class="txt" href="#E12:35_1_1_24_4_3_4_1_1_4"><span class="lab"><font color="Green" title="E136">A964</font></span></a>, <a class="ref" href="gobrd13.html#T45" title="GOBRD13:th.45">GOBRD13:45</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E17:35_1_1_24_4_3_4_1_1_4"/><span class="lab"><font color="Green" title="E139">A967</font></span>: 
<font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">p2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:35_1_1_24_4_3_4_1_1_4"><span class="lab"><font color="Green" title="E137">A965</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E18:35_1_1_24_4_3_4_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E140">A968</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">p1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c24">p2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E121">A942</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E122">A943</font></span></a>, <a class="txt" href="#E16:35_1_1_24_4_3_4_1_1_4"><span class="lab"><font color="Green" title="E138">A966</font></span></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></span>;<br/><a NAME="E19:35_1_1_24_4_3_4_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E141">A969</font></span>: 
<font color="Maroon" title="c23">p1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E20:35_1_1_24_4_3_4_1_1_4"/>
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">p1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c24">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:35_1_1_24_4"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4"><span class="lab"><font color="Green" title="E91">A826</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E3:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E121">A942</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E122">A943</font></span></a>, <a class="txt" href="#E17:35_1_1_24_4_3_4_1_1_4"><span class="lab"><font color="Green" title="E139">A967</font></span></a>, <a class="txt" href="#E16:35_1_1_24_4_3_4_1_1_4"><span class="lab"><font color="Green" title="E138">A966</font></span></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></span>;<br/><a NAME="E21:35_1_1_24_4_3_4_1_1_4"/><span class="kw">then </span>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c23">p1</font>,<font color="Maroon" title="c24">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_1_1_24_4"><span class="lab"><font color="Green" title="E98">A833</font></span></a>, <a class="txt" href="#E13:35_1_1_24_4"><span class="lab"><font color="Green" title="E96">A831</font></span></a>, <a class="txt" href="#E15:35_1_1_24_4"><span class="lab"><font color="Green" title="E97">A832</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4"><span class="lab"><font color="Green" title="E99">A834</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4"><span class="lab"><font color="Green" title="E100">A835</font></span></a>, <a class="txt" href="#E18:35_1_1_24_4_3_4_1_1_4"><span class="lab"><font color="Green" title="E140">A968</font></span></a>, <a class="txt" href="#E5:35_1_1_24_4_3_4_1_1_4"><span class="lab"><font color="Green" title="E129">A957</font></span></a>, <a class="txt" href="#E6:35_1_1_24_4_3_4_1_1_4"><span class="lab"><font color="Green" title="E130">A958</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>;<br/><a NAME="E22:35_1_1_24_4_3_4_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:35_1_1_24_4"><span class="lab"><font color="Green" title="E104">A839</font></span></a>, <a class="txt" href="#E26:35_1_1_24_4"><span class="lab"><font color="Green" title="E107">A842</font></span></a>, <a class="txt" href="#E4:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E122">A943</font></span></a>, <a class="txt" href="#E7:35_1_1_24_4_3_4"><span class="lab"><font color="Green" title="E125">A946</font></span></a>, <a class="txt" href="#E7:35_1_1_24_4_3_4_1_1_4"><span class="lab"><font color="Green" title="E131">A959</font></span></a>, <a class="txt" href="#E13:35_1_1_24_4_3_4_1_1_4"><span class="lab"><font color="Green" title="E137">A965</font></span></a>, <a class="txt" href="#E19:35_1_1_24_4_3_4_1_1_4"><span class="lab"><font color="Green" title="E141">A969</font></span></a>, <a class="ref" href="jordan9.html#T22" target="_self" title="JORDAN9:th.22">Th22</a></span>;<br/><a NAME="E23:35_1_1_24_4_3_4_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E142">A970</font></span>: 
<font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:35_1_1_24_4"><span class="lab"><font color="Green" title="E119">A854</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E24:35_1_1_24_4_3_4_1_1_4"/>
( <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_24"><span class="lab"><font color="Green" title="E85">A791</font></span></a>, <a class="txt" href="#E16:35_1_1_24_4"><span class="lab"><font color="Green" title="E98">A833</font></span></a>, <a class="txt" href="#E13:35_1_1_24_4"><span class="lab"><font color="Green" title="E96">A831</font></span></a>, <a class="txt" href="#E15:35_1_1_24_4"><span class="lab"><font color="Green" title="E97">A832</font></span></a>, <a class="ref" href="jordan9.html#T29" target="_self" title="JORDAN9:th.29">Th29</a>, <a class="ref" href="sppol_2.html#T22" title="SPPOL_2:th.22">SPPOL_2:22</a></span>;<br/><span class="kw">hence </span><a NAME="E25:35_1_1_24_4_3_4_1_1_4"/>
<font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c17">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E63">A572</font></span></a>, <a class="txt" href="#E1:35_1_1_24"><span class="lab"><font color="Green" title="E83">A773</font></span></a>, <a class="txt" href="#E37:35_1_1_24_4"><span class="lab"><font color="Green" title="E115">A850</font></span></a>, <a class="txt" href="#E38:35_1_1_24_4"><span class="lab"><font color="Green" title="E116">A851</font></span></a>, <a class="txt" href="#E44:35_1_1_24_4"><span class="lab"><font color="Green" title="E119">A854</font></span></a>, <a class="txt" href="#E23:35_1_1_24_4_3_4_1_1_4"><span class="lab"><font color="Green" title="E142">A970</font></span></a>, <a class="txt" href="#E2:35_1_1_24_4_3_4_1_1_4"><span class="lab"><font color="Green" title="E127">A955</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E11:35_1_1_24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A971</font></span>: 
(  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c17">g</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c17">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard9.html#T27" title="GOBOARD9:th.27">GOBOARD9:27</a>, <a class="ref" href="goboard9.html#D1" title="GOBOARD9:def.1">GOBOARD9:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">Lg9</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E13:35_1_1_24"/><span class="lab"><font color="Green" title="E90">A972</font></span>: 
 <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c17">g</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard9.html#D2" title="GOBOARD9:def.2">GOBOARD9:def 2</a></span>;<br/><a NAME="E14:35_1_1_24"/><span class="lab"><font color="Green" title="E91">A973</font></span>: 
<font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">Lg9</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_24_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c19">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> or <font color="Maroon" title="c19">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Lg9</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_1_24_5"/><span class="lab"><font color="Green" title="E92">A974</font></span>: 
<font color="Maroon" title="c19">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">and </span><br/><a NAME="E2:35_1_1_24_5"/><span class="lab"><font color="Green" title="E93">A975</font></span>: 
not <font color="Maroon" title="c19">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Lg9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c20">c</font> = <font color="Maroon" title="c19">c</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_24_5"><span class="lab"><font color="Green" title="E92">A974</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c21">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:35_1_1_24_5"/><span class="lab"><font color="Green" title="E94">A976</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">i</font>
 <span class="kw">and </span><br/><a NAME="E6:35_1_1_24_5"/><span class="lab"><font color="Green" title="E95">A977</font></span>: 
<font color="Maroon" title="c21">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font>
 <span class="kw">and </span><br/><a NAME="E7:35_1_1_24_5"/><span class="lab"><font color="Green" title="E96">A978</font></span>: 
<font color="Maroon" title="c20">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_24_5"><span class="lab"><font color="Green" title="E93">A975</font></span></a>, <a class="ref" href="sppol_2.html#T14" title="SPPOL_2:th.14">SPPOL_2:14</a>, <a class="ref" href="subset_1.html#T50" title="SUBSET_1:th.50">SUBSET_1:50</a></span>;<br/>
<a NAME="E8:35_1_1_24_5"/><span class="lab"><font color="Green" title="E97">A979</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_24_5"><span class="lab"><font color="Green" title="E94">A976</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E9:35_1_1_24_5"/>
<font color="Maroon" title="c21">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_24_5"><span class="lab"><font color="Green" title="E94">A976</font></span></a>, <a class="txt" href="#E6:35_1_1_24_5"><span class="lab"><font color="Green" title="E95">A977</font></span></a>, <a class="ref" href="seq_4.html#T151" title="SEQ_4:th.151">SEQ_4:151</a></span>;<br/>
<a NAME="E10:35_1_1_24_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A980</font></span>: 
<font color="Maroon" title="c17">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></span>;<br/>
<a NAME="E11:35_1_1_24_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E12:35_1_1_24_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A981</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1_24_5"><span class="lab"><font color="Green" title="E95">A977</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E13:35_1_1_24_5"/>
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_24_5"><span class="lab"><font color="Green" title="E94">A976</font></span></a>, <a class="txt" href="#E6:35_1_1_24_5"><span class="lab"><font color="Green" title="E95">A977</font></span></a>, <a class="ref" href="seq_4.html#T151" title="SEQ_4:th.151">SEQ_4:151</a></span>;<br/>
<a NAME="E14:35_1_1_24_5"/><span class="kw">then </span>
<font color="Maroon" title="c17">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></span>;<br/>
<a NAME="E15:35_1_1_24_5"/><span class="kw">then </span>
<font color="Maroon" title="c20">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c14">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:35_1_1"><span class="lab"><font color="Green" title="E69">A578</font></span></a>, <a class="txt" href="#E7:35_1_1_24_5"><span class="lab"><font color="Green" title="E96">A978</font></span></a>, <a class="txt" href="#E10:35_1_1_24_5"><span class="lab"><font color="Green" title="E98">A980</font></span></a>, <a class="txt" href="#E8:35_1_1_24_5"><span class="lab"><font color="Green" title="E97">A979</font></span></a>, <a class="txt" href="#E12:35_1_1_24_5"><span class="lab"><font color="Green" title="E99">A981</font></span></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></span>;<br/>
<a NAME="E16:35_1_1_24_5"/><span class="kw">then </span>
<font color="Maroon" title="c20">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E52:35_1_1"><span class="lab"><font color="Green" title="E69">A578</font></span></a>, <a class="txt" href="#E8:35_1_1_24_5"><span class="lab"><font color="Green" title="E97">A979</font></span></a>, <a class="txt" href="#E12:35_1_1_24_5"><span class="lab"><font color="Green" title="E99">A981</font></span></a>, <a class="ref" href="gobrd13.html#T30" title="GOBRD13:th.30">GOBRD13:30</a></span>;<br/>
<a NAME="E17:35_1_1_24_5"/><span class="kw">then </span>
<font color="Maroon" title="c20">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E18:35_1_1_24_5"/><span class="kw">then </span>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_24_5"><span class="lab"><font color="Green" title="E92">A974</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:35_1_1_24_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E52:35_1_1"><span class="lab"><font color="Green" title="E69">A578</font></span></a>, <a class="txt" href="#E8:35_1_1_24_5"><span class="lab"><font color="Green" title="E97">A979</font></span></a>, <a class="txt" href="#E12:35_1_1_24_5"><span class="lab"><font color="Green" title="E99">A981</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E15:35_1_1_24"/><span class="lab"><font color="Green" title="E92">A982</font></span>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#D8" title="EUCLID:def.8">EUCLID:def 8</a></span>;<br/><a NAME="E16:35_1_1_24"/>
<font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c17">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_24_6"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_1_1_24_6"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a>, <a class="txt" href="#E38:35_1_1"><span class="lab"><font color="Green" title="E61">A570</font></span></a>, <a class="txt" href="#E48:35_1_1"><span class="lab"><font color="Green" title="E67">A576</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:35_1_1_24_6"/><span class="lab"><font color="Green" title="E93">A983</font></span>: 
<font color="Maroon" title="c19">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1_24_6"/><span class="lab"><font color="Green" title="E94">A984</font></span>: 
<font color="Maroon" title="c19">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">p</font> = <font color="Maroon" title="c19">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_24_6"><span class="lab"><font color="Green" title="E93">A983</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_24_6_1"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c21">u</font> = <font color="Maroon" title="c20">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c22">a</font> = <font color="Maroon" title="c20">p</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c17">g</font> )</span><br/><span class="kw">thus </span><a NAME="E2:35_1_1_24_6_1"/>
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_24_6"><span class="lab"><font color="Green" title="E94">A984</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c17">g</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c23">r</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E4:35_1_1_24_6_1"/><span class="lab"><font color="Green" title="E95">A985</font></span>: 
<font color="Maroon" title="c23">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E5:35_1_1_24_6_1"/><span class="lab"><font color="Green" title="E96">A986</font></span>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c21">u</font>,<font color="Maroon" title="c23">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:35_1_1_24"><span class="lab"><font color="Green" title="E91">A973</font></span></a>, <a class="txt" href="#E4:35_1_1_24"><span class="lab"><font color="Green" title="E84">A790</font></span></a>, <a class="txt" href="#E4:35_1_1_24_6"><span class="lab"><font color="Green" title="E94">A984</font></span></a>, <a class="ref" href="goboard6.html#T8" title="GOBOARD6:th.8">GOBOARD6:8</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c24">r</font> = <font color="Maroon" title="c23">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><a NAME="E7:35_1_1_24_6_1"/><span class="lab"><font color="Green" title="E97">A987</font></span>: 
<font color="Maroon" title="c20">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c21">u</font>,<font color="Maroon" title="c24">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_24_6_1"><span class="lab"><font color="Green" title="E95">A985</font></span></a>, <a class="ref" href="goboard6.html#T4" title="GOBOARD6:th.4">GOBOARD6:4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c25">B</font> =  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c21">u</font>,<font color="Maroon" title="c24">r</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1_24"><span class="lab"><font color="Green" title="E92">A982</font></span></a>, <a class="txt" href="#E4:35_1_1_24_6_1"><span class="lab"><font color="Green" title="E95">A985</font></span></a>, <a class="ref" href="goboard6.html#T4" title="GOBOARD6:th.4">GOBOARD6:4</a>, <a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></span>;<br/><a NAME="E9:35_1_1_24_6_1"/><span class="lab"><font color="Green" title="E98">A988</font></span>: 
<font color="Maroon" title="c20">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_24_6_1"><span class="lab"><font color="Green" title="E95">A985</font></span></a>, <a class="ref" href="goboard6.html#T4" title="GOBOARD6:th.4">GOBOARD6:4</a></span>;<br/><a NAME="E10:35_1_1_24_6_1"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c12">m</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c25">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:35_1_1"><span class="lab"><font color="Green" title="E61">A570</font></span></a>, <a class="txt" href="#E48:35_1_1"><span class="lab"><font color="Green" title="E67">A576</font></span></a>, <a class="txt" href="#E72:35_1_1"><span class="lab"><font color="Green" title="E82">A756</font></span></a>, <a class="ref" href="goboard6.html#T6" title="GOBOARD6:th.6">GOBOARD6:6</a></span>;<br/><a NAME="E11:35_1_1_24_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A989</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c12">m</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c25">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_24_6"><span class="lab"><font color="Green" title="E93">A983</font></span></a>, <a class="txt" href="#E7:35_1_1_24_6_1"><span class="lab"><font color="Green" title="E97">A987</font></span></a>, <a class="ref" href="tops_1.html#T39" title="TOPS_1:th.39">TOPS_1:39</a></span>;<br/><a NAME="E12:35_1_1_24_6_1"/>
(  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c17">g</font>,1,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Maroon" title="c17">g</font>,1</span>)</span> &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Maroon" title="c17">g</font>,1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c17">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:35_1_1"><span class="lab"><font color="Green" title="E78">A625</font></span></a>, <a class="txt" href="#E65:35_1_1"><span class="lab"><font color="Green" title="E77">A624</font></span></a>, <a class="ref" href="goboard9.html#T28" title="GOBOARD9:th.28">GOBOARD9:28</a>, <a class="ref" href="gobrd13.html#T34" title="GOBRD13:th.34">GOBRD13:34</a>, <a class="ref" href="tops_1.html#T48" title="TOPS_1:th.48">TOPS_1:48</a></span>;<br/><a NAME="E13:35_1_1_24_6_1"/><span class="kw">then </span>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c17">g</font>,1,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c17">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E14:35_1_1_24_6_1"/><span class="kw">then </span>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c17">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:35_1_1"><span class="lab"><font color="Green" title="E62">A571</font></span></a>, <a class="txt" href="#E51:35_1_1"><span class="lab"><font color="Green" title="E68">A577</font></span></a>, <a class="txt" href="#E65:35_1_1"><span class="lab"><font color="Green" title="E77">A624</font></span></a>, <a class="ref" href="gobrd13.html#T33" title="GOBRD13:th.33">GOBRD13:33</a></span>;<br/><a NAME="E15:35_1_1_24_6_1"/><span class="kw">then </span>
( <font color="Maroon" title="c25">B</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c14">f</font>,<font color="Maroon" title="c12">m</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c17">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:35_1_1"><span class="lab"><font color="Green" title="E61">A570</font></span></a>, <a class="ref" href="sprect_3.html#T17" title="SPRECT_3:th.17">SPRECT_3:17</a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/><a NAME="E16:35_1_1_24_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c25">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c17">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:35_1_1_24"><span class="lab"><font color="Green" title="E90">A972</font></span></a>, <a class="txt" href="#E5:35_1_1_24_6_1"><span class="lab"><font color="Green" title="E96">A986</font></span></a>, <a class="txt" href="#E11:35_1_1_24_6_1"><span class="lab"><font color="Green" title="E99">A989</font></span></a>, <a class="ref" href="goboard9.html#T6" title="GOBOARD9:th.6">GOBOARD9:6</a></span>;<br/><span class="kw">hence </span><a NAME="E17:35_1_1_24_6_1"/>
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c17">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_24_6_1"><span class="lab"><font color="Green" title="E98">A988</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:35_1_1_24_6"/>
<font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c17">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E17:35_1_1_24"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E13:35_1_1_24"><span class="lab"><font color="Green" title="E90">A972</font></span></a>, <a class="txt" href="#E14:35_1_1_24"><span class="lab"><font color="Green" title="E91">A973</font></span></a>, <a class="txt" href="#E11:35_1_1_24"><span class="lab"><font color="Green" title="E89">A971</font></span></a>, <a class="ref" href="jordan9.html#T3" target="_self" title="JORDAN9:th.3">Th3</a>, <a class="ref" href="sprect_4.html#T7" title="SPRECT_4:th.7">SPRECT_4:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><font color="Maroon" title="c17">g</font> = 
<font color="Maroon" title="c14">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T234" title="XREAL_1:th.234">XREAL_1:234</a></span>
<br/>.= 
<font color="Maroon" title="c14">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T31" title="FINSEQ_5:th.31">FINSEQ_5:31</a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">f</font> = <font color="Maroon" title="c14">f</font> as  non  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>   <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a>  <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a> ;<br/>
<a NAME="E76:35_1_1"/>
<font color="Maroon" title="c18">f</font> is  <a href="sprect_2.html#V1" title="SPRECT_2:attr.1">clockwise_oriented</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_26"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_1_1_26"/>
<font color="Maroon" title="c18">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T69" title="RLTOPSP1:th.69">RLTOPSP1:69</a></span>;<br/>
<a NAME="E2:35_1_1_26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A990</font></span>: 
<font color="Maroon" title="c18">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:35_1_1"><span class="lab"><font color="Green" title="E65">A574</font></span></a>, <a class="ref" href="sppol_2.html#T15" title="SPPOL_2:th.15">SPPOL_2:15</a></span>;<br/>
<a NAME="E3:35_1_1_26"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">NW-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">f</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>;<br/>
<a NAME="E4:35_1_1_26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A991</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">NW-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">f</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_26"><span class="lab"><font color="Green" title="E83">A990</font></span></a>, <a class="ref" href="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></span>;<br/>
<a NAME="E5:35_1_1_26"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:35_1_1"><span class="lab"><font color="Green" title="E44">A231</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E6:35_1_1_26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A992</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:35_1_1_26"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">NE-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">f</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>;<br/>
<a NAME="E8:35_1_1_26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A993</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">NE-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">f</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_26"><span class="lab"><font color="Green" title="E83">A990</font></span></a>, <a class="ref" href="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></span>;<br/>
<a NAME="E9:35_1_1_26"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">f</font> holds <br/>(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c18">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c18">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a></span>;<br/>
<a NAME="E10:35_1_1_26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A994</font></span>: 
 <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:35_1_1"><span class="lab"><font color="Green" title="E62">A571</font></span></a>, <a class="txt" href="#E59:35_1_1"><span class="lab"><font color="Green" title="E73">A608</font></span></a>, <a class="ref" href="jordan9.html#T32" target="_self" title="JORDAN9:th.32">Th32</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E12:35_1_1_26"/><span class="lab"><font color="Green" title="E88">A995</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i</font>
 <span class="kw">and </span><br/><a NAME="E13:35_1_1_26"/><span class="lab"><font color="Green" title="E89">A996</font></span>: 
<font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E14:35_1_1_26"/><span class="lab"><font color="Green" title="E90">A997</font></span>: 
( <font color="Maroon" title="c18">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c18">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E15:35_1_1_26"/>
 <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c19">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E16:35_1_1_26"/>
 <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:35_1_1"><span class="lab"><font color="Green" title="E73">A608</font></span></a></span>;<br/>
<a NAME="E17:35_1_1_26"/>
<font color="Maroon" title="c19">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:35_1_1_26"><span class="lab"><font color="Green" title="E89">A996</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E18:35_1_1_26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A998</font></span>: 
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">f</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E6:35_1_1_26"><span class="lab"><font color="Green" title="E85">A992</font></span></a>, <a class="txt" href="#E12:35_1_1_26"><span class="lab"><font color="Green" title="E88">A995</font></span></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/>
<a NAME="E19:35_1_1_26"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E20:35_1_1_26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A999</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E6:35_1_1_26"><span class="lab"><font color="Green" title="E85">A992</font></span></a>, <a class="txt" href="#E13:35_1_1_26"><span class="lab"><font color="Green" title="E89">A996</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/>
<a NAME="E21:35_1_1_26"/>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">NW-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">f</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">NE-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">f</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">NE-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">f</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">f</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a>, <a class="ref" href="pscomp_1.html#T82" title="PSCOMP_1:th.82">PSCOMP_1:82</a></span>;<br/>
<a NAME="E22:35_1_1_26"/><span class="kw">then </span>
<font color="Maroon" title="c18">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">NW-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">NE-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:35_1_1_26"><span class="lab"><font color="Green" title="E87">A994</font></span></a>, <a class="txt" href="#E14:35_1_1_26"><span class="lab"><font color="Green" title="E90">A997</font></span></a>, <a class="txt" href="#E18:35_1_1_26"><span class="lab"><font color="Green" title="E91">A998</font></span></a>, <a class="txt" href="#E20:35_1_1_26"><span class="lab"><font color="Green" title="E92">A999</font></span></a>, <a class="txt" href="#E4:35_1_1_26"><span class="lab"><font color="Green" title="E84">A991</font></span></a>, <a class="txt" href="#E8:35_1_1_26"><span class="lab"><font color="Green" title="E86">A993</font></span></a>, <a class="ref" href="goboard7.html#T9" title="GOBOARD7:th.9">GOBOARD7:9</a></span>;<br/>
<a NAME="E23:35_1_1_26"/><span class="kw">then </span>
<font color="Maroon" title="c18">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">NW-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">f</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">NE-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">f</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_26"><span class="lab"><font color="Green" title="E83">A990</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:35_1_1_26"/>
<font color="Maroon" title="c18">f</font> is  <a href="sprect_2.html#V1" title="SPRECT_2:attr.1">clockwise_oriented</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:35_1_1_26"><span class="lab"><font color="Green" title="E87">A994</font></span></a>, <a class="ref" href="sprect_2.html#T34" title="SPRECT_2:th.34">SPRECT_2:34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">f</font> = <font color="Maroon" title="c18">f</font> as  non  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>   <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a>   <a href="sprect_2.html#V1" title="SPRECT_2:attr.1">clockwise_oriented</a>  <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c19">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp;  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c19">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c19">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">f</font> holds <br/>( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c19">f</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Olive" title="b1">k</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c19">f</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Olive" title="b1">k</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c19">f</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Olive" title="b1">k</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E78:35_1_1"/>
<font color="Maroon" title="c19">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E47">A428</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c19">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c19">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">f</font> holds <br/>( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c19">f</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Olive" title="b1">k</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c19">f</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Olive" title="b1">k</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c19">f</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Olive" title="b1">k</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E79:35_1_1"/>
 ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c19">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c19">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:35_1_1"><span class="lab"><font color="Green" title="E73">A608</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">f</font> holds <br/>( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c19">f</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Olive" title="b1">k</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c19">f</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Olive" title="b1">k</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c19">f</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Olive" title="b1">k</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c20">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">m</font> &amp; <font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">f</font> implies ( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Maroon" title="c20">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Maroon" title="c20">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c19">f</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c20">m</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Maroon" title="c20">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Maroon" title="c20">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c19">f</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Maroon" title="c20">m</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Maroon" title="c20">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c19">f</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c20">m</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E80:35_1_1"/><span class="lab"><font color="Green" title="E83">A1000</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">m</font>
 <span class="kw">and </span><br/><a NAME="E81:35_1_1"/><span class="lab"><font color="Green" title="E84">A1001</font></span>: 
<font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Maroon" title="c20">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Maroon" title="c20">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c19">f</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c20">m</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Maroon" title="c20">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Maroon" title="c20">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c19">f</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Maroon" title="c20">m</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Maroon" title="c20">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c19">f</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c20">m</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) )</span><br/>

<a NAME="E82:35_1_1"/><span class="lab"><font color="Green" title="E85">A1002</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:35_1_1"><span class="lab"><font color="Green" title="E53">A490</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E81:35_1_1"><span class="lab"><font color="Green" title="E84">A1001</font></span></a></span>;<br/>
<a NAME="E83:35_1_1"/><span class="lab"><font color="Green" title="E86">A1003</font></span>: 
<font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E84:35_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A1004</font></span>: 
<font color="Maroon" title="c19">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:35_1_1"><span class="lab"><font color="Green" title="E53">A490</font></span></a>, <a class="txt" href="#E57:35_1_1"><span class="lab"><font color="Green" title="E71">A580</font></span></a>, <a class="txt" href="#E81:35_1_1"><span class="lab"><font color="Green" title="E84">A1001</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E85:35_1_1"/><span class="lab"><font color="Green" title="E88">A1005</font></span>: 
<font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E81:35_1_1"><span class="lab"><font color="Green" title="E84">A1001</font></span></a>, <a class="txt" href="#E83:35_1_1"><span class="lab"><font color="Green" title="E86">A1003</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E86:35_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A1006</font></span>: 
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c20">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Maroon" title="c20">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:35_1_1"><span class="lab"><font color="Green" title="E62">A571</font></span></a>, <a class="txt" href="#E80:35_1_1"><span class="lab"><font color="Green" title="E83">A1000</font></span></a>, <a class="txt" href="#E84:35_1_1"><span class="lab"><font color="Green" title="E87">A1004</font></span></a>, <a class="ref" href="gobrd13.html#T43" title="GOBRD13:th.43">GOBRD13:43</a></span>;<br/>
<a NAME="E87:35_1_1"/><span class="lab"><font color="Green" title="E90">A1007</font></span>: 
( <font color="Maroon" title="c20">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 &amp; <font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:35_1_1"><span class="lab"><font color="Green" title="E83">A1000</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/>
<a NAME="E88:35_1_1"/><span class="lab"><font color="Green" title="E91">A1008</font></span>: 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c20">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Maroon" title="c20">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:35_1_1"><span class="lab"><font color="Green" title="E62">A571</font></span></a>, <a class="txt" href="#E80:35_1_1"><span class="lab"><font color="Green" title="E83">A1000</font></span></a>, <a class="txt" href="#E85:35_1_1"><span class="lab"><font color="Green" title="E88">A1005</font></span></a>, <a class="txt" href="#E84:35_1_1"><span class="lab"><font color="Green" title="E87">A1004</font></span></a>, <a class="ref" href="gobrd13.html#T43" title="GOBRD13:th.43">GOBRD13:43</a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Maroon" title="c20">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Maroon" title="c20">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c19">f</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Maroon" title="c20">m</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Maroon" title="c20">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c19">f</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c20">m</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_27"/>
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Maroon" title="c20">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Maroon" title="c20">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">f</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c20">m</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><a NAME="E2:35_1_1_27"/><span class="kw">then </span>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c20">m</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_1_1"><span class="lab"><font color="Green" title="E49">A458</font></span></a>, <a class="txt" href="#E87:35_1_1"><span class="lab"><font color="Green" title="E90">A1007</font></span></a>, <a class="txt" href="#E86:35_1_1"><span class="lab"><font color="Green" title="E89">A1006</font></span></a>, <a class="txt" href="#E88:35_1_1"><span class="lab"><font color="Green" title="E91">A1008</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_27"/>
<font color="Maroon" title="c19">f</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c20">m</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:35_1_1"><span class="lab"><font color="Green" title="E83">A1000</font></span></a>, <a class="txt" href="#E81:35_1_1"><span class="lab"><font color="Green" title="E84">A1001</font></span></a>, <a class="txt" href="#E82:35_1_1"><span class="lab"><font color="Green" title="E85">A1002</font></span></a>, <a class="ref" href="gobrd13.html#T44" title="GOBRD13:th.44">GOBRD13:44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Maroon" title="c20">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c19">f</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c20">m</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_28"/>
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Maroon" title="c20">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Maroon" title="c20">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">f</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Maroon" title="c20">m</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/><a NAME="E2:35_1_1_28"/><span class="kw">then </span>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Maroon" title="c20">m</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_1_1"><span class="lab"><font color="Green" title="E49">A458</font></span></a>, <a class="txt" href="#E87:35_1_1"><span class="lab"><font color="Green" title="E90">A1007</font></span></a>, <a class="txt" href="#E86:35_1_1"><span class="lab"><font color="Green" title="E89">A1006</font></span></a>, <a class="txt" href="#E88:35_1_1"><span class="lab"><font color="Green" title="E91">A1008</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1_1_28"/>
<font color="Maroon" title="c19">f</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Maroon" title="c20">m</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:35_1_1"><span class="lab"><font color="Green" title="E83">A1000</font></span></a>, <a class="txt" href="#E81:35_1_1"><span class="lab"><font color="Green" title="E84">A1001</font></span></a>, <a class="txt" href="#E82:35_1_1"><span class="lab"><font color="Green" title="E85">A1002</font></span></a>, <a class="ref" href="gobrd13.html#T46" title="GOBRD13:th.46">GOBRD13:46</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E91:35_1_1"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c19">f</font>,<font color="Maroon" title="c20">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">f</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c20">m</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span><br/>

<a NAME="E92:35_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c20">m</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_1_1"><span class="lab"><font color="Green" title="E49">A458</font></span></a>, <a class="txt" href="#E87:35_1_1"><span class="lab"><font color="Green" title="E90">A1007</font></span></a>, <a class="txt" href="#E86:35_1_1"><span class="lab"><font color="Green" title="E89">A1006</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E93:35_1_1"/>
<font color="Maroon" title="c19">f</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c20">m</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:35_1_1"><span class="lab"><font color="Green" title="E83">A1000</font></span></a>, <a class="txt" href="#E81:35_1_1"><span class="lab"><font color="Green" title="E84">A1001</font></span></a>, <a class="txt" href="#E82:35_1_1"><span class="lab"><font color="Green" title="E85">A1002</font></span></a>, <a class="ref" href="gobrd13.html#T45" title="GOBRD13:th.45">GOBRD13:45</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
