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

<b>set </b><font color="Maroon" title="c3">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/>
<b>set </b><font color="Maroon" title="c4">W</font> =  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">C</font>;<br/>
<b>set </b><font color="Maroon" title="c5">E</font> =  <a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <font color="Maroon" title="c1">C</font>;<br/>
<b>set </b><font color="Maroon" title="c6">S</font> =  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font>;<br/>
<b>set </b><font color="Maroon" title="c7">N</font> =  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">C</font>;<br/>
<a NAME="E1:33_1_1"/><i><font color="Green" title="E31">A2</font></i>: 
(  <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#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3 &amp;  <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> )
 
<b>by </b><i><a class="ref" href="jordan8.html#D1" title="JORDAN8:def.1">JORDAN8:def 1</a></i>;<br/>
<b>defpred </b><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> ] <b>means </b>( ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 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#K1" title="NAT_1:func.1">+</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#K29" title="PSCOMP_1:func.29">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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#K29" title="PSCOMP_1:func.29">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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="binarith.html#K3" title="BINARITH:func.3">-'</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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="binarith.html#K3" title="BINARITH:func.3">-'</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="E2:33_1_1"/><i><font color="Green" title="E32">A3</font></i>: 
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="33_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><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> ;<br/><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>consider </b><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> <b> such that </b><br/><a NAME="E2:33_1_1_1"/><i><font color="Green" title="E33">A4</font></i>: 
( 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#K1" title="NAT_1:func.1">+</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> )
 <b>and </b><br/><a NAME="E3:33_1_1_1"/><i><font color="Green" title="E34">A5</font></i>: 
(  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="jordan9.html#T30" target="_self" title="JORDAN9:th.30">Th30</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1_1"/>
( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 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><b>by </b><i><a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_1"/><i><font color="Green" title="E35">A6</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>take </b>
<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>
;<br/><b>thus </b><a NAME="E2:33_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>]
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_1"><i><font color="Green" title="E35">A6</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_2"/><i><font color="Green" title="E35">A7</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>take </b>
<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#K1" title="NAT_1:func.1">+</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>
;<br/><b>thus </b><a NAME="E2:33_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#K1" title="NAT_1:func.1">+</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>]
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_2"><i><font color="Green" title="E35">A7</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_1_1_3"/><i><font color="Green" title="E35">A8</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>and </b><br/><a NAME="E2:33_1_1_1_1_3"/><i><font color="Green" title="E36">A9</font></i>: 
<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>
 ;<br/><div class="add"><b>reconsider </b><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> <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_1_1_3"><i><font color="Green" title="E36">A9</font></i></a></i>;<br/><b>thus </b><a NAME="E4:33_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>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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="33_1_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1"/><i><font color="Green" title="E37">A10</font></i>: 
 <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>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:33_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>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="33_1_1_1_1_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E38">A11</font></i>: 
( <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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/><div class="add"><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E39">A12</font></i>: 
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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a></i>;<br/><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E40">A13</font></i>: 
<span 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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><b>then consider </b><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> <b> such that </b><br/><a NAME="E5:33_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E41">A14</font></i>: 
( <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> &amp; <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4: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> <font color="Maroon" title="c11">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>and </b><br/><a NAME="E6:33_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E42">A15</font></i>: 
( <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> &amp; <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 <b>and </b><br/><a NAME="E7:33_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E43">A16</font></i>: 
( ( <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E8:33_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E44">A17</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i1</font> &amp; <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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j1</font> &amp; <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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E9:33_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E45">A18</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i2</font> &amp; <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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j2</font> &amp; <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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E10:33_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E46">A19</font></i>: 
( <font color="Maroon" title="c12">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c13">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E8:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E11:33_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E47">A20</font></i>: 
( 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#K1" title="NAT_1:func.1">+</a> 1 &amp; 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#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E12:33_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E48">A21</font></i>: 
( <font color="Maroon" title="c14">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c15">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E13:33_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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="ref" href="binarith.html#T52" title="BINARITH:th.52">BINARITH:52</a></i>;<br/><a NAME="E14:33_1_1_1_1_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E49">A22</font></i>: 
( <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="binarith.html#K3" title="BINARITH:func.3">-'</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> &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> <font color="Maroon" title="c11">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E15:33_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E50">A23</font></i>: 
<span 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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E40">A13</font></i></a></i>;<br/><b>thus </b><a NAME="E16:33_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>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E51">A24</font></i>: 
(  <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:33_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>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E43">A16</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E52">A25</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> )
 ;<br/><div class="add"><b>take </b><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#K1" title="NAT_1:func.1">+</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>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><b>take </b><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c15">j2</font>;<br/><b>thus </b><a NAME="E1:33_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="binarith.html#K3" title="BINARITH:func.3">-'</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="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><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> 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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E53">A26</font></i>: 
( <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> &amp; <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> )
 <b>and </b><br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E54">A27</font></i>: 
( <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4: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> <font color="Maroon" title="c11">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">i1'</font>,<font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/>

<a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4: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> <font color="Maroon" title="c11">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">i1'</font>,<font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E14:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E49">A22</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E54">A27</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E55">A28</font></i>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i1'</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2'</font> &amp; <font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2'</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E53">A26</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E43">A16</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E55">A28</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2'</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_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#K1" title="NAT_1:func.1">+</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>
 ;<br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 &amp; <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
 <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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E52">A25</font></i></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="jordan8.html#T19" title="JORDAN8:th.19">JORDAN8:19</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_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">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<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>
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E47">A20</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E55">A28</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E4:33_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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j2'</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E50">A23</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E55">A28</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:33_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">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E52">A25</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E55">A28</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_3"><b>case </b></a><a NAME="E1:33_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">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E52">A25</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E55">A28</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_4"><b>case </b></a><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<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> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E52">A25</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E55">A28</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_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>]
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E51">A24</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E52">A29</font></i>: 
( <font color="Maroon" title="c12">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>take </b><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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c14">i2</font>;<br/><b>take </b><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c15">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><b>thus </b><a NAME="E1:33_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="binarith.html#K3" title="BINARITH:func.3">-'</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="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><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> 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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><i><font color="Green" title="E53">A30</font></i>: 
( <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> &amp; <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> )
 <b>and </b><br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><i><font color="Green" title="E54">A31</font></i>: 
( <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4: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> <font color="Maroon" title="c11">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">i1'</font>,<font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/>

<a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4: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> <font color="Maroon" title="c11">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">i1'</font>,<font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E14:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E49">A22</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><i><font color="Green" title="E54">A31</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E55">A32</font></i>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i1'</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2'</font> &amp; <font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2'</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><i><font color="Green" title="E53">A30</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E43">A16</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><i><font color="Green" title="E55">A32</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<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> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><i><font color="Green" title="E55">A32</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:33_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">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ;<br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c15">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c15">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54" title="BINARITH:th.54">BINARITH:54</a></i>;<br/><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2_1"/><b>then </b>
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E52">A29</font></i></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></i>;<br/><a NAME="E6:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2_1"/><b>then </b>
 <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>,0 <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E8:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="jordan8.html#T20" title="JORDAN8:th.20">JORDAN8:20</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_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">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E48">A21</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><i><font color="Green" title="E55">A32</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E4:33_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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E50">A23</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><i><font color="Green" title="E55">A32</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_3"><b>case </b></a><a NAME="E1:33_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">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><i><font color="Green" title="E55">A32</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_4"><b>case </b></a><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<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> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><i><font color="Green" title="E55">A32</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_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>]
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E51">A24</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"/><i><font color="Green" title="E52">A33</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>take </b><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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c14">i2</font>;<br/><b>take </b><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><b>thus </b><a NAME="E1:33_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="binarith.html#K3" title="BINARITH:func.3">-'</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="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><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> 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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><i><font color="Green" title="E53">A34</font></i>: 
( <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> &amp; <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> )
 <b>and </b><br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><i><font color="Green" title="E54">A35</font></i>: 
( <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4: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> <font color="Maroon" title="c11">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">i1'</font>,<font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/>

<a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/>
( <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4: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> <font color="Maroon" title="c11">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">i1'</font>,<font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E14:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E49">A22</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><i><font color="Green" title="E54">A35</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><b>then </b><i><font color="Green" title="E55">A36</font></i>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i1'</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2'</font> &amp; <font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2'</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><i><font color="Green" title="E53">A34</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E43">A16</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><i><font color="Green" title="E55">A36</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_1"><b>case </b></a><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<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> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E52">A33</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><i><font color="Green" title="E55">A36</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_2"><b>case </b></a><a NAME="E1:33_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">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E52">A33</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><i><font color="Green" title="E55">A36</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3"><b>case </b></a><a NAME="E1:33_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">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_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#K1" title="NAT_1:func.1">+</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>
 ;<br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3_1"/><b>then </b>
( <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 &amp; <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3_1"/><b>then </b>
 <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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E52">A33</font></i></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="jordan8.html#T18" title="JORDAN8:th.18">JORDAN8:18</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_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">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E47">A20</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><i><font color="Green" title="E55">A36</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E4:33_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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E50">A23</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><i><font color="Green" title="E55">A36</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_4"><b>case </b></a><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<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> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E52">A33</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><i><font color="Green" title="E55">A36</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_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>]
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E51">A24</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"/><i><font color="Green" title="E52">A37</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><b>take </b><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="binarith.html#K3" title="BINARITH:func.3">-'</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>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c14">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><b>take </b><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c15">j2</font>;<br/><b>thus </b><a NAME="E1:33_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="binarith.html#K3" title="BINARITH:func.3">-'</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="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><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> 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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><i><font color="Green" title="E53">A38</font></i>: 
( <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> &amp; <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> )
 <b>and </b><br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><i><font color="Green" title="E54">A39</font></i>: 
( <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4: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> <font color="Maroon" title="c11">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">i1'</font>,<font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/>

<a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/>
( <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4: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> <font color="Maroon" title="c11">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">i1'</font>,<font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E14:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E49">A22</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><i><font color="Green" title="E54">A39</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><b>then </b><i><font color="Green" title="E55">A40</font></i>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i1'</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2'</font> &amp; <font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2'</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><i><font color="Green" title="E53">A38</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E43">A16</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><i><font color="Green" title="E55">A40</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_1"><b>case </b></a><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<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> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"><i><font color="Green" title="E52">A37</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><i><font color="Green" title="E55">A40</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_2"><b>case </b></a><a NAME="E1:33_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">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"><i><font color="Green" title="E52">A37</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><i><font color="Green" title="E55">A40</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_3"><b>case </b></a><a NAME="E1:33_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">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"><i><font color="Green" title="E52">A37</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><i><font color="Green" title="E55">A40</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4"><b>case </b></a><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ;<br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c14">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c14">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54" title="BINARITH:th.54">BINARITH:54</a></i>;<br/><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4_1"/><b>then </b>
 <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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"><i><font color="Green" title="E52">A37</font></i></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></i>;<br/><a NAME="E6:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4_1"/><b>then </b>
 <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>,0,<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>
 <b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="jordan8.html#T21" title="JORDAN8:th.21">JORDAN8:21</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_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">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<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>
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E48">A21</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><i><font color="Green" title="E55">A40</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E4:33_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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2'</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E50">A23</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><i><font color="Green" title="E55">A40</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_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>]
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E51">A24</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E51">A41</font></i>: 
(  <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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/><div class="add"><b>thus </b><a NAME="E2:33_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>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E43">A16</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E52">A42</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> )
 ;<br/><div class="add"><b>take </b><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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c14">i2</font>;<br/><b>take </b><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><b>thus </b><a NAME="E1:33_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="binarith.html#K3" title="BINARITH:func.3">-'</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="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><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> 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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><i><font color="Green" title="E53">A43</font></i>: 
( <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> &amp; <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> )
 <b>and </b><br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><i><font color="Green" title="E54">A44</font></i>: 
( <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4: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> <font color="Maroon" title="c11">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">i1'</font>,<font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/>

<a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/>
( <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4: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> <font color="Maroon" title="c11">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">i1'</font>,<font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E14:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E49">A22</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><i><font color="Green" title="E54">A44</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E55">A45</font></i>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i1'</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2'</font> &amp; <font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2'</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><i><font color="Green" title="E53">A43</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E43">A16</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><i><font color="Green" title="E55">A45</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2'</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_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#K1" title="NAT_1:func.1">+</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>
 ;<br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 &amp; <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1_1"/><b>then </b>
 <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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E51">A41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E52">A42</font></i></a>, <a class="ref" href="gobrd13.html#T36" title="GOBRD13:th.36">GOBRD13:36</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E8:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="jordan8.html#T18" title="JORDAN8:th.18">JORDAN8:18</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_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">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E47">A20</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><i><font color="Green" title="E55">A45</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E4:33_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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E50">A23</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><i><font color="Green" title="E55">A45</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:33_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">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<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> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E52">A42</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><i><font color="Green" title="E55">A45</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_3"><b>case </b></a><a NAME="E1:33_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">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<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> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E52">A42</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><i><font color="Green" title="E55">A45</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_4"><b>case </b></a><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E52">A42</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><i><font color="Green" title="E55">A45</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_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>]
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E51">A41</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E52">A46</font></i>: 
( <font color="Maroon" title="c12">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>take </b><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#K1" title="NAT_1:func.1">+</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>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><b>take </b><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c15">j2</font>;<br/><b>thus </b><a NAME="E1:33_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="binarith.html#K3" title="BINARITH:func.3">-'</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="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><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> 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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><i><font color="Green" title="E53">A47</font></i>: 
( <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> &amp; <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> )
 <b>and </b><br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><i><font color="Green" title="E54">A48</font></i>: 
( <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4: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> <font color="Maroon" title="c11">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">i1'</font>,<font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/>

<a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4: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> <font color="Maroon" title="c11">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">i1'</font>,<font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E14:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E49">A22</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E54">A48</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E55">A49</font></i>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i1'</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2'</font> &amp; <font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2'</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E53">A47</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E43">A16</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E55">A49</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E52">A46</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E55">A49</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:33_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">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_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#K1" title="NAT_1:func.1">+</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>
 ;<br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 &amp; <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2_1"/><b>then </b>
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E51">A41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E52">A46</font></i></a>, <a class="ref" href="gobrd13.html#T38" title="GOBRD13:th.38">GOBRD13:38</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E46">A19</font></i></a>, <a class="ref" href="jordan8.html#T19" title="JORDAN8:th.19">JORDAN8:19</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_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">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<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>
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E47">A20</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E55">A49</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E4:33_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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j2'</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E50">A23</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E55">A49</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_3"><b>case </b></a><a NAME="E1:33_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">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<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> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E52">A46</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E55">A49</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_4"><b>case </b></a><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E52">A46</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E55">A49</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_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>]
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E51">A41</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"/><i><font color="Green" title="E52">A50</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>take </b><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="binarith.html#K3" title="BINARITH:func.3">-'</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>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c14">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><b>take </b><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c15">j2</font>;<br/><b>thus </b><a NAME="E1:33_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="binarith.html#K3" title="BINARITH:func.3">-'</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="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><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> 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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><i><font color="Green" title="E53">A51</font></i>: 
( <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> &amp; <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> )
 <b>and </b><br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><i><font color="Green" title="E54">A52</font></i>: 
( <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4: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> <font color="Maroon" title="c11">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">i1'</font>,<font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/>

<a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/>
( <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4: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> <font color="Maroon" title="c11">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">i1'</font>,<font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E14:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E49">A22</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><i><font color="Green" title="E54">A52</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><b>then </b><i><font color="Green" title="E55">A53</font></i>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i1'</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2'</font> &amp; <font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2'</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E43">A16</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><i><font color="Green" title="E55">A53</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_1"><b>case </b></a><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"><i><font color="Green" title="E52">A50</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><i><font color="Green" title="E55">A53</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_2"><b>case </b></a><a NAME="E1:33_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">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<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> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"><i><font color="Green" title="E52">A50</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><i><font color="Green" title="E55">A53</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3"><b>case </b></a><a NAME="E1:33_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">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ;<br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c14">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c14">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54" title="BINARITH:th.54">BINARITH:54</a></i>;<br/><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3_1"/><b>then </b>
 <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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E51">A41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"><i><font color="Green" title="E52">A50</font></i></a>, <a class="ref" href="gobrd13.html#T40" title="GOBRD13:th.40">GOBRD13:40</a></i>;<br/><a NAME="E6:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3_1"/><b>then </b>
 <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>,0,<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>
 <b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="jordan8.html#T21" title="JORDAN8:th.21">JORDAN8:21</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_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">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<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>
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E48">A21</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><i><font color="Green" title="E55">A53</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E4:33_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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2'</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E50">A23</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><i><font color="Green" title="E55">A53</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_4"><b>case </b></a><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"><i><font color="Green" title="E52">A50</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><i><font color="Green" title="E55">A53</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_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>]
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E51">A41</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"/><i><font color="Green" title="E52">A54</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><b>take </b><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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c14">i2</font>;<br/><b>take </b><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c15">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><b>thus </b><a NAME="E1:33_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="binarith.html#K3" title="BINARITH:func.3">-'</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="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><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> 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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><i><font color="Green" title="E53">A55</font></i>: 
( <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> &amp; <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> )
 <b>and </b><br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><i><font color="Green" title="E54">A56</font></i>: 
( <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4: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> <font color="Maroon" title="c11">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">i1'</font>,<font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/>

<a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/>
( <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4: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> <font color="Maroon" title="c11">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">i1'</font>,<font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E14:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E49">A22</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><i><font color="Green" title="E54">A56</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><b>then </b><i><font color="Green" title="E55">A57</font></i>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i1'</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2'</font> &amp; <font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2'</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><i><font color="Green" title="E53">A55</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E43">A16</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><i><font color="Green" title="E55">A57</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_1"><b>case </b></a><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"><i><font color="Green" title="E52">A54</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><i><font color="Green" title="E55">A57</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_2"><b>case </b></a><a NAME="E1:33_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">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<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> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"><i><font color="Green" title="E52">A54</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><i><font color="Green" title="E55">A57</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_3"><b>case </b></a><a NAME="E1:33_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">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<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> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"><i><font color="Green" title="E52">A54</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><i><font color="Green" title="E55">A57</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4"><b>case </b></a><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ;<br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c15">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c15">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54" title="BINARITH:th.54">BINARITH:54</a></i>;<br/><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4_1"/><b>then </b>
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E51">A41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"><i><font color="Green" title="E52">A54</font></i></a>, <a class="ref" href="gobrd13.html#T42" title="GOBRD13:th.42">GOBRD13:42</a></i>;<br/><a NAME="E6:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4_1"/><b>then </b>
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,0 <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E10:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E46">A19</font></i></a>, <a class="ref" href="jordan8.html#T20" title="JORDAN8:th.20">JORDAN8:20</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_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">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E48">A21</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><i><font color="Green" title="E55">A57</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E4:33_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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E50">A23</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><i><font color="Green" title="E55">A57</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_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>]
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E51">A41</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3"/><i><font color="Green" title="E51">A58</font></i>: 
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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/><div class="add"><b>thus </b><a NAME="E2:33_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>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E43">A16</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"/><i><font color="Green" title="E52">A59</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> )
 ;<br/><div class="add"><b>take </b><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="binarith.html#K3" title="BINARITH:func.3">-'</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>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c14">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><b>take </b><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c15">j2</font>;<br/><b>thus </b><a NAME="E1:33_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="binarith.html#K3" title="BINARITH:func.3">-'</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="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><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> 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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><i><font color="Green" title="E53">A60</font></i>: 
( <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> &amp; <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> )
 <b>and </b><br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><i><font color="Green" title="E54">A61</font></i>: 
( <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4: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> <font color="Maroon" title="c11">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">i1'</font>,<font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/>

<a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/>
( <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4: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> <font color="Maroon" title="c11">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">i1'</font>,<font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E14:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E49">A22</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><i><font color="Green" title="E54">A61</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E55">A62</font></i>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i1'</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2'</font> &amp; <font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2'</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><i><font color="Green" title="E53">A60</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E43">A16</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><i><font color="Green" title="E55">A62</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2'</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ;<br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54" title="BINARITH:th.54">BINARITH:54</a></i>;<br/><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1_1"/><b>then </b>
 <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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E51">A58</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E52">A59</font></i></a>, <a class="ref" href="gobrd13.html#T35" title="GOBRD13:th.35">GOBRD13:35</a></i>;<br/><a NAME="E6:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1_1"/><b>then </b>
 <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>,0,<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>
 <b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="jordan8.html#T21" title="JORDAN8:th.21">JORDAN8:21</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_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">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<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>
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E48">A21</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><i><font color="Green" title="E55">A62</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E4:33_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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2'</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E50">A23</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><i><font color="Green" title="E55">A62</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:33_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">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E52">A59</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><i><font color="Green" title="E55">A62</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_3"><b>case </b></a><a NAME="E1:33_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">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E52">A59</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><i><font color="Green" title="E55">A62</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_4"><b>case </b></a><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<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> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E52">A59</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><i><font color="Green" title="E55">A62</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_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>]
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E51">A58</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"/><i><font color="Green" title="E52">A63</font></i>: 
( <font color="Maroon" title="c12">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>take </b><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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c14">i2</font>;<br/><b>take </b><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><b>thus </b><a NAME="E1:33_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="binarith.html#K3" title="BINARITH:func.3">-'</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="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><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> 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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><i><font color="Green" title="E53">A64</font></i>: 
( <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> &amp; <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> )
 <b>and </b><br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><i><font color="Green" title="E54">A65</font></i>: 
( <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4: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> <font color="Maroon" title="c11">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">i1'</font>,<font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/>

<a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/>
( <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4: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> <font color="Maroon" title="c11">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">i1'</font>,<font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E14:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E49">A22</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E54">A65</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E55">A66</font></i>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i1'</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2'</font> &amp; <font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2'</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E53">A64</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E43">A16</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E55">A66</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<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> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"><i><font color="Green" title="E52">A63</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E55">A66</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:33_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">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_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#K1" title="NAT_1:func.1">+</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>
 ;<br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 &amp; <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2_1"/><b>then </b>
 <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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E51">A58</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"><i><font color="Green" title="E52">A63</font></i></a>, <a class="ref" href="gobrd13.html#T37" title="GOBRD13:th.37">GOBRD13:37</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="jordan8.html#T18" title="JORDAN8:th.18">JORDAN8:18</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_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">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E47">A20</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E55">A66</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E4:33_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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E50">A23</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E55">A66</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_3"><b>case </b></a><a NAME="E1:33_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">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"><i><font color="Green" title="E52">A63</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E55">A66</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_4"><b>case </b></a><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<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> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"><i><font color="Green" title="E52">A63</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E55">A66</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_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>]
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E51">A58</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"/><i><font color="Green" title="E52">A67</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>take </b><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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c14">i2</font>;<br/><b>take </b><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c15">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><b>thus </b><a NAME="E1:33_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="binarith.html#K3" title="BINARITH:func.3">-'</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="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><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> 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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><i><font color="Green" title="E53">A68</font></i>: 
( <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> &amp; <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> )
 <b>and </b><br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><i><font color="Green" title="E54">A69</font></i>: 
( <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4: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> <font color="Maroon" title="c11">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">i1'</font>,<font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/>

<a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/>
( <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4: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> <font color="Maroon" title="c11">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">i1'</font>,<font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E14:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E49">A22</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><i><font color="Green" title="E54">A69</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><b>then </b><i><font color="Green" title="E55">A70</font></i>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i1'</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2'</font> &amp; <font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2'</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><i><font color="Green" title="E53">A68</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E43">A16</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><i><font color="Green" title="E55">A70</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_1"><b>case </b></a><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<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> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"><i><font color="Green" title="E52">A67</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><i><font color="Green" title="E55">A70</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_2"><b>case </b></a><a NAME="E1:33_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">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"><i><font color="Green" title="E52">A67</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><i><font color="Green" title="E55">A70</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3"><b>case </b></a><a NAME="E1:33_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">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ;<br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c15">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c15">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54" title="BINARITH:th.54">BINARITH:54</a></i>;<br/><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3_1"/><b>then </b>
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E51">A58</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"><i><font color="Green" title="E52">A67</font></i></a>, <a class="ref" href="gobrd13.html#T39" title="GOBRD13:th.39">GOBRD13:39</a></i>;<br/><a NAME="E6:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3_1"/><b>then </b>
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,0 <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E12:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E48">A21</font></i></a>, <a class="ref" href="jordan8.html#T20" title="JORDAN8:th.20">JORDAN8:20</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_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">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E48">A21</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><i><font color="Green" title="E55">A70</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E4:33_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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E50">A23</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><i><font color="Green" title="E55">A70</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_4"><b>case </b></a><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<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> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"><i><font color="Green" title="E52">A67</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><i><font color="Green" title="E55">A70</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_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>]
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E51">A58</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"/><i><font color="Green" title="E52">A71</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><b>take </b><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#K1" title="NAT_1:func.1">+</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>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><b>take </b><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c15">j2</font>;<br/><b>thus </b><a NAME="E1:33_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="binarith.html#K3" title="BINARITH:func.3">-'</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="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><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> 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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><i><font color="Green" title="E53">A72</font></i>: 
( <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> &amp; <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> )
 <b>and </b><br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><i><font color="Green" title="E54">A73</font></i>: 
( <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4: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> <font color="Maroon" title="c11">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">i1'</font>,<font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/>

<a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/>
( <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4: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> <font color="Maroon" title="c11">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">i1'</font>,<font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E14:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E49">A22</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><i><font color="Green" title="E54">A73</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><b>then </b><i><font color="Green" title="E55">A74</font></i>: 
( <font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i1'</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1'</font> &amp; <font color="Maroon" title="c14">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2'</font> &amp; <font color="Maroon" title="c15">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2'</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><i><font color="Green" title="E53">A72</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E43">A16</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><i><font color="Green" title="E55">A74</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_1"><b>case </b></a><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<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> &amp; <font color="Maroon" title="c16">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"><i><font color="Green" title="E52">A71</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><i><font color="Green" title="E55">A74</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_2"><b>case </b></a><a NAME="E1:33_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">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"><i><font color="Green" title="E52">A71</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><i><font color="Green" title="E55">A74</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_3"><b>case </b></a><a NAME="E1:33_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">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"><i><font color="Green" title="E52">A71</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><i><font color="Green" title="E55">A74</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4"><b>case </b></a><a NAME="E1:33_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">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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_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#K1" title="NAT_1:func.1">+</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>
 ;<br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4_1"/><b>then </b>
( <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 &amp; <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4_1"/><b>then </b>
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E51">A58</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"><i><font color="Green" title="E52">A71</font></i></a>, <a class="ref" href="gobrd13.html#T41" title="GOBRD13:th.41">GOBRD13:41</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E48">A21</font></i></a>, <a class="ref" href="jordan8.html#T19" title="JORDAN8:th.19">JORDAN8:19</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_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">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<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>
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E47">A20</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><i><font color="Green" title="E55">A74</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E4:33_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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j2'</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E50">A23</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><i><font color="Green" title="E55">A74</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_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>]
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E51">A58</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_2"/><i><font color="Green" title="E38">A75</font></i>: 
(  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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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> )
 ;<br/><div class="add"><b>take </b>
<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>
;<br/><b>thus </b><a NAME="E2:33_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>]
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_1_1_1_2"><i><font color="Green" title="E38">A75</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_2"/><i><font color="Green" title="E37">A76</font></i>: 
 <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>
 ;<br/><div class="add"><b>take </b>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
;<br/><b>thus </b><a NAME="E2:33_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> ]
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3_1_1_2"><i><font color="Green" title="E37">A76</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_4"/><i><font color="Green" title="E35">A77</font></i>: 
( <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> )
 ;<br/><div class="add"><b>take </b>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
;<br/><b>thus </b><a NAME="E2:33_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> ]
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_4"><i><font color="Green" title="E35">A77</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c8">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:33_1_1"/><i><font color="Green" title="E33">A78</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">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> 
 <b>and </b><br/><a NAME="E5:33_1_1"/><i><font color="Green" title="E34">A79</font></i>: 
<font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E6:33_1_1"/><i><font color="Green" title="E35">A80</font></i>: 
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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>]
 <b>from </b><i><a class="ref" href="lattice5.html#S1" title="LATTICE5:sch.1">LATTICE5:sch 1</a>(<a class="txt" href="jordan9.html#E2:33_1_1"><i><font color="Green" title="E32">A3</font></i></a>);<br/></i>
<b>defpred </b><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> ] <b>means </b><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="E7:33_1_1"/><i><font color="Green" title="E36">A81</font></i>: 
 <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="E8:33_1_1"/><b>then </b><i><font color="Green" title="E37">A82</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1"><i><font color="Green" title="E34">A79</font></i></a></i>;<br/>
<a NAME="E9:33_1_1"/><i><font color="Green" title="E38">A83</font></i>: 
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#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><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> ;<br/>

<b>assume </b><a NAME="E1:33_1_1_2"/><i><font color="Green" title="E39">A84</font></i>: 
<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>
 ;<br/>

<a NAME="E2:33_1_1_2"/><i><font color="Green" title="E40">A85</font></i>: 
<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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>]
 
<b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1"><i><font color="Green" title="E35">A80</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_2_1"/>
( <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 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><b>by </b><i><a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_2_1_1"/>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_2_1_1"/>
<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#K1" title="NAT_1:func.1">+</a> 1</span>)</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_2"><i><font color="Green" title="E40">A85</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_2_1_2"/>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_2_1_2"/>
<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#K1" title="NAT_1:func.1">+</a> 1</span>)</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_2"><i><font color="Green" title="E40">A85</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_2_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_2_1_3"/><i><font color="Green" title="E41">A86</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><div class="add"><b>thus </b><a NAME="E2:33_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#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_2_1_3_1"><b>proof </b></a><div class="add">

<b>reconsider </b><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> <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_2"><i><font color="Green" title="E39">A84</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_2_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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="33_1_1_2_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_2_1_3_1_1_1"/><i><font color="Green" title="E42">A87</font></i>: 
 <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>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:33_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#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_2_1_3_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_2_1_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="33_1_1_2_1_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_2_1_3_1_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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/><div class="add"><a NAME="E2:33_1_1_2_1_3_1_1_1_1_1_1"/><b>then </b>
( (  <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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> ) ) &amp; (  <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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> ) ) &amp; (  <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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> ) ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1"><i><font color="Green" title="E35">A80</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_2_1_3"><i><font color="Green" title="E41">A86</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_2_1_3_1_1_1"><i><font color="Green" title="E42">A87</font></i></a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_2_1_3_1_1_1_1_1_1"/>
<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#K1" title="NAT_1:func.1">+</a> 1</span>)</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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_2_1_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_2_1_3_1_1_1_1_1_2"/><i><font color="Green" title="E43">A88</font></i>: 
(  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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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> )
 ;<br/><div class="add"><a NAME="E2:33_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/><b>hence </b><a NAME="E3:33_1_1_2_1_3_1_1_1_1_1_2"/>
<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#K1" title="NAT_1:func.1">+</a> 1</span>)</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1"><i><font color="Green" title="E35">A80</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_2_1_3"><i><font color="Green" title="E41">A86</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_2_1_3_1_1_1"><i><font color="Green" title="E42">A87</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_2_1_3_1_1_1_1_1_2"><i><font color="Green" title="E43">A88</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_2_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:33_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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_2_1_3_1_1_2"/>
<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#K1" title="NAT_1:func.1">+</a> 1</span>)</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1"><i><font color="Green" title="E35">A80</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1"><i><font color="Green" title="E36">A81</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_2_1_3"><i><font color="Green" title="E41">A86</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E10:33_1_1"/><i><font color="Green" title="E39">A89</font></i>: 
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>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="jordan9.html#E8:33_1_1"><i><font color="Green" title="E37">A82</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1"><i><font color="Green" title="E38">A83</font></i></a>);<br/></i>
<a NAME="E11:33_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="33_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:33_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>
 ;<br/>

<a NAME="E2:33_1_1_3"/><b>then </b>
 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#K1" title="RELAT_1:func.1">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> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E3:33_1_1_3"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1"><i><font color="Green" title="E33">A78</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1"><i><font color="Green" title="E39">A89</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:33_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> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><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> ,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>  <b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1"><i><font color="Green" title="E33">A78</font></i></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></i>;<br/>
<b>defpred </b><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> ] <b>means </b> <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="E13:33_1_1"/><i><font color="Green" title="E40">A90</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1"><i><font color="Green" title="E34">A79</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E14:33_1_1"/><i><font color="Green" title="E41">A91</font></i>: 
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#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><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> ;<br/>

<b>assume </b><a NAME="E1:33_1_1_4"/><i><font color="Green" title="E42">A92</font></i>: 
 <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>
 ;<br/>

<a NAME="E2:33_1_1_4"/><i><font color="Green" title="E43">A93</font></i>: 
<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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>]
 
<b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1"><i><font color="Green" title="E35">A80</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_4_1"/>
( <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 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><b>by </b><i><a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_4_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_4_1_1"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_4"><i><font color="Green" title="E43">A93</font></i></a>, <a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_4_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_4_1_2"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_4"><i><font color="Green" title="E43">A93</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_4_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_4_1_3"/><i><font color="Green" title="E44">A94</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><div class="add"><b>thus </b><a NAME="E2:33_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#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_4_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_4_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="33_1_1_4_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_4_1_3_1_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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/><div class="add"><a NAME="E2:33_1_1_4_1_3_1_1_1"/><b>then </b>
( (  <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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> ) ) &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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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> ) ) &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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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> ) ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1"><i><font color="Green" title="E35">A80</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_4"><i><font color="Green" title="E42">A92</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_4_1_3"><i><font color="Green" title="E44">A94</font></i></a></i>;<br/><b>hence </b><a NAME="E3:33_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#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_4"><i><font color="Green" title="E42">A92</font></i></a>, <a class="ref" href="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_4_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:33_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_4_1_3_1_1_2"/><b>then </b>
<font 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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1"><i><font color="Green" title="E35">A80</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_4"><i><font color="Green" title="E42">A92</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_4_1_3"><i><font color="Green" title="E44">A94</font></i></a></i>;<br/><b>hence </b><a NAME="E3:33_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#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_4"><i><font color="Green" title="E42">A92</font></i></a>, <a class="ref" href="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E15:33_1_1"/><i><font color="Green" title="E42">A95</font></i>: 
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>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="jordan9.html#E13:33_1_1"><i><font color="Green" title="E40">A90</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1"><i><font color="Green" title="E41">A91</font></i></a>);<br/></i>
<b>defpred </b><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> ] <b>means </b>( <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#K1" title="NAT_1:func.1">+</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="E16:33_1_1"/><i><font color="Green" title="E43">A96</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_5"><b>proof </b></a><div class="add">

<a NAME="E1:33_1_1_5"/>
( ( 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> 0</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> 0</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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> 0</span>)</span> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> 0</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> 0</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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> 0</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K3" title="REAL_1:func.3">+</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 ) )
 
<b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1"><i><font color="Green" title="E34">A79</font></i></a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>;<br/>
<b>hence </b><a NAME="E2:33_1_1_5"/>
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> 0 <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>
 <b>by </b><i><a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/>

<b>let </b><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> ;<br/>

<b>assume </b><a NAME="E3:33_1_1_5"/>
( 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#K1" title="NAT_1:func.1">+</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> 0</span>)</span> )
 ;<br/>

<b>hence </b><a NAME="E4:33_1_1_5"/>
(  <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> 0</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> 0</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1"><i><font color="Green" title="E34">A79</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E44">A97</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_6"><b>now </b></a><div class="add"><b>let </b><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> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_6"/><i><font color="Green" title="E45">A98</font></i>: 
<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>
 <b>and </b><br/><a NAME="E2:33_1_1_6"/><i><font color="Green" title="E46">A99</font></i>: 
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#K1" title="NAT_1:func.1">+</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> )
 <b>and </b><br/><a NAME="E3:33_1_1_6"/><i><font color="Green" title="E47">A100</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><b>let </b><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> ;<br/><b>assume </b><b>that </b><br/><a NAME="E4:33_1_1_6"/><i><font color="Green" title="E48">A101</font></i>: 
( <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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>and </b><br/><a NAME="E5:33_1_1_6"/><i><font color="Green" title="E49">A102</font></i>: 
( <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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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> )
 ;<br/><a NAME="E6:33_1_1_6"/><i><font color="Green" title="E50">A103</font></i>: 
 <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>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a></i>;<br/><a NAME="E7:33_1_1_6"/><b>then </b><i><font color="Green" title="E51">A104</font></i>: 
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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_6"><i><font color="Green" title="E47">A100</font></i></a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a></i>;<br/><a NAME="E8:33_1_1_6"/><i><font color="Green" title="E52">A105</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_6"><i><font color="Green" title="E47">A100</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_6"><i><font color="Green" title="E50">A103</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><a NAME="E9:33_1_1_6"/><b>then </b><i><font color="Green" title="E53">A106</font></i>: 
(  <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_6"><i><font color="Green" title="E46">A99</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_6"><i><font color="Green" title="E51">A104</font></i></a></i>;<br/><a NAME="E10:33_1_1_6"/><i><font color="Green" title="E54">A107</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i1</font> &amp; <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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j1</font> &amp; <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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_6"><i><font color="Green" title="E48">A101</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E11:33_1_1_6"/><i><font color="Green" title="E55">A108</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">i2</font> &amp; <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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j2</font> &amp; <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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1_6"><i><font color="Green" title="E49">A102</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E12:33_1_1_6"/><i><font color="Green" title="E56">A109</font></i>: 
( <font color="Maroon" title="c11">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c12">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E10:33_1_1_6"><i><font color="Green" title="E54">A107</font></i></a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E13:33_1_1_6"/><i><font color="Green" title="E57">A110</font></i>: 
( 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#K1" title="NAT_1:func.1">+</a> 1 &amp; 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#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E14:33_1_1_6"/><i><font color="Green" title="E58">A111</font></i>: 
( <font color="Maroon" title="c13">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:33_1_1_6_1"/><i><font color="Green" title="E59">A112</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_6_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_6_1_1"/>
<font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 ;<br/><a NAME="E2:33_1_1_6_1_1"/><b>then </b>
( <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 &amp; <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:33_1_1_6_1_1"/><b>then </b>
<font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_6_1_1"/><b>then </b>
 <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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_6"><i><font color="Green" title="E45">A98</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_6"><i><font color="Green" title="E48">A101</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_6"><i><font color="Green" title="E49">A102</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_6"><i><font color="Green" title="E51">A104</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_6"><i><font color="Green" title="E52">A105</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_6"><i><font color="Green" title="E53">A106</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_6_1"><i><font color="Green" title="E59">A112</font></i></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_6_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_6"><i><font color="Green" title="E54">A107</font></i></a>, <a class="ref" href="jordan8.html#T19" title="JORDAN8:th.19">JORDAN8:19</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_6_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_6"><i><font color="Green" title="E57">A110</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:33_1_1_6_2"/><i><font color="Green" title="E59">A113</font></i>: 
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_6_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_6_2_1"/>
<font color="Maroon" title="c14">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ;<br/><a NAME="E2:33_1_1_6_2_1"/><b>then </b>
<font color="Maroon" title="c14">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E3:33_1_1_6_2_1"/><b>then </b>
<font color="Maroon" title="c14">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54" title="BINARITH:th.54">BINARITH:54</a></i>;<br/><a NAME="E4:33_1_1_6_2_1"/><b>then </b>
<font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:33_1_1_6_2_1"/><b>then </b>
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_6"><i><font color="Green" title="E45">A98</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_6"><i><font color="Green" title="E48">A101</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_6"><i><font color="Green" title="E49">A102</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_6"><i><font color="Green" title="E51">A104</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_6"><i><font color="Green" title="E52">A105</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_6"><i><font color="Green" title="E53">A106</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_6_2"><i><font color="Green" title="E59">A113</font></i></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></i>;<br/><a NAME="E6:33_1_1_6_2_1"/><b>then </b>
 <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>,0 <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_6_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E10:33_1_1_6"><i><font color="Green" title="E54">A107</font></i></a>, <a class="ref" href="jordan8.html#T20" title="JORDAN8:th.20">JORDAN8:20</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_6_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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_6"><i><font color="Green" title="E58">A111</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:33_1_1_6_3"/><i><font color="Green" title="E59">A114</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_6_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_6_3_1"/>
<font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 ;<br/><a NAME="E2:33_1_1_6_3_1"/><b>then </b>
( <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 &amp; <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:33_1_1_6_3_1"/><b>then </b>
<font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_6_3_1"/><b>then </b>
 <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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_6"><i><font color="Green" title="E45">A98</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_6"><i><font color="Green" title="E48">A101</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_6"><i><font color="Green" title="E49">A102</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_6"><i><font color="Green" title="E51">A104</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_6"><i><font color="Green" title="E52">A105</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_6"><i><font color="Green" title="E53">A106</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_6_3"><i><font color="Green" title="E59">A114</font></i></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_6_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="ref" href="jordan8.html#T18" title="JORDAN8:th.18">JORDAN8:18</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_6_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_6"><i><font color="Green" title="E57">A110</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:33_1_1_6_4"/><i><font color="Green" title="E59">A115</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_6_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_6_4_1"/>
<font color="Maroon" title="c13">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ;<br/><a NAME="E2:33_1_1_6_4_1"/><b>then </b>
<font color="Maroon" title="c13">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E3:33_1_1_6_4_1"/><b>then </b>
<font color="Maroon" title="c13">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54" title="BINARITH:th.54">BINARITH:54</a></i>;<br/><a NAME="E4:33_1_1_6_4_1"/><b>then </b>
<font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:33_1_1_6_4_1"/><b>then </b>
 <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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_6"><i><font color="Green" title="E45">A98</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_6"><i><font color="Green" title="E48">A101</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_6"><i><font color="Green" title="E49">A102</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_6"><i><font color="Green" title="E51">A104</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_6"><i><font color="Green" title="E52">A105</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_6"><i><font color="Green" title="E53">A106</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_6_4"><i><font color="Green" title="E59">A115</font></i></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></i>;<br/><a NAME="E6:33_1_1_6_4_1"/><b>then </b>
 <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>,0,<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>
 <b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_6_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="ref" href="jordan8.html#T21" title="JORDAN8:th.21">JORDAN8:21</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_6_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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_6"><i><font color="Green" title="E58">A111</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_6_5"/><i><font color="Green" title="E59">A116</font></i>: 
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E2:33_1_1_6_5"/><i><font color="Green" title="E60">A117</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_6_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_6_5_1"/>
<font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 ;<br/><a NAME="E2:33_1_1_6_5_1"/><b>then </b>
( <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 &amp; <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:33_1_1_6_5_1"/><b>then </b>
<font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_6_5_1"/><b>then </b>
 <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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_6"><i><font color="Green" title="E45">A98</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_6"><i><font color="Green" title="E48">A101</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_6"><i><font color="Green" title="E49">A102</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_6"><i><font color="Green" title="E51">A104</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_6"><i><font color="Green" title="E52">A105</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_6_5"><i><font color="Green" title="E59">A116</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_6_5"><i><font color="Green" title="E60">A117</font></i></a>, <a class="ref" href="gobrd13.html#T36" title="GOBRD13:th.36">GOBRD13:36</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_6_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E10:33_1_1_6"><i><font color="Green" title="E54">A107</font></i></a>, <a class="ref" href="jordan8.html#T18" title="JORDAN8:th.18">JORDAN8:18</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_6_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_6"><i><font color="Green" title="E57">A110</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_6_6"/><i><font color="Green" title="E59">A118</font></i>: 
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E2:33_1_1_6_6"/><i><font color="Green" title="E60">A119</font></i>: 
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_6_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_6_6_1"/>
<font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 ;<br/><a NAME="E2:33_1_1_6_6_1"/><b>then </b>
( <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 &amp; <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:33_1_1_6_6_1"/><b>then </b>
<font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_6_6_1"/><b>then </b>
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_6"><i><font color="Green" title="E45">A98</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_6"><i><font color="Green" title="E48">A101</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_6"><i><font color="Green" title="E49">A102</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_6"><i><font color="Green" title="E51">A104</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_6"><i><font color="Green" title="E52">A105</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_6_6"><i><font color="Green" title="E59">A118</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_6_6"><i><font color="Green" title="E60">A119</font></i></a>, <a class="ref" href="gobrd13.html#T38" title="GOBRD13:th.38">GOBRD13:38</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_6_6_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_6"><i><font color="Green" title="E56">A109</font></i></a>, <a class="ref" href="jordan8.html#T19" title="JORDAN8:th.19">JORDAN8:19</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_6_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_6"><i><font color="Green" title="E57">A110</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_6_7"/><i><font color="Green" title="E59">A120</font></i>: 
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E2:33_1_1_6_7"/><i><font color="Green" title="E60">A121</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_6_7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_6_7_1"/>
<font color="Maroon" title="c13">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ;<br/><a NAME="E2:33_1_1_6_7_1"/><b>then </b>
<font color="Maroon" title="c13">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E3:33_1_1_6_7_1"/><b>then </b>
<font color="Maroon" title="c13">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54" title="BINARITH:th.54">BINARITH:54</a></i>;<br/><a NAME="E4:33_1_1_6_7_1"/><b>then </b>
<font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:33_1_1_6_7_1"/><b>then </b>
 <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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_6"><i><font color="Green" title="E45">A98</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_6"><i><font color="Green" title="E48">A101</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_6"><i><font color="Green" title="E49">A102</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_6"><i><font color="Green" title="E51">A104</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_6"><i><font color="Green" title="E52">A105</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_6_7"><i><font color="Green" title="E59">A120</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_6_7"><i><font color="Green" title="E60">A121</font></i></a>, <a class="ref" href="gobrd13.html#T40" title="GOBRD13:th.40">GOBRD13:40</a></i>;<br/><a NAME="E6:33_1_1_6_7_1"/><b>then </b>
 <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>,0,<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>
 <b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_6_7_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_6"><i><font color="Green" title="E54">A107</font></i></a>, <a class="ref" href="jordan8.html#T21" title="JORDAN8:th.21">JORDAN8:21</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_6_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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_6"><i><font color="Green" title="E58">A111</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_6_8"/><i><font color="Green" title="E59">A122</font></i>: 
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E2:33_1_1_6_8"/><i><font color="Green" title="E60">A123</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_6_8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_6_8_1"/>
<font color="Maroon" title="c14">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ;<br/><a NAME="E2:33_1_1_6_8_1"/><b>then </b>
<font color="Maroon" title="c14">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E3:33_1_1_6_8_1"/><b>then </b>
<font color="Maroon" title="c14">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54" title="BINARITH:th.54">BINARITH:54</a></i>;<br/><a NAME="E4:33_1_1_6_8_1"/><b>then </b>
<font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:33_1_1_6_8_1"/><b>then </b>
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_6"><i><font color="Green" title="E45">A98</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_6"><i><font color="Green" title="E48">A101</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_6"><i><font color="Green" title="E49">A102</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_6"><i><font color="Green" title="E51">A104</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_6"><i><font color="Green" title="E52">A105</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_6_8"><i><font color="Green" title="E59">A122</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_6_8"><i><font color="Green" title="E60">A123</font></i></a>, <a class="ref" href="gobrd13.html#T42" title="GOBRD13:th.42">GOBRD13:42</a></i>;<br/><a NAME="E6:33_1_1_6_8_1"/><b>then </b>
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,0 <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_6_8_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E12:33_1_1_6"><i><font color="Green" title="E56">A109</font></i></a>, <a class="ref" href="jordan8.html#T20" title="JORDAN8:th.20">JORDAN8:20</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_6_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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_6"><i><font color="Green" title="E58">A111</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_6_9"/><i><font color="Green" title="E59">A124</font></i>: 
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E2:33_1_1_6_9"/><i><font color="Green" title="E60">A125</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_6_9_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_6_9_1"/>
<font color="Maroon" title="c13">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ;<br/><a NAME="E2:33_1_1_6_9_1"/><b>then </b>
<font color="Maroon" title="c13">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E3:33_1_1_6_9_1"/><b>then </b>
<font color="Maroon" title="c13">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54" title="BINARITH:th.54">BINARITH:54</a></i>;<br/><a NAME="E4:33_1_1_6_9_1"/><b>then </b>
<font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:33_1_1_6_9_1"/><b>then </b>
 <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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_6"><i><font color="Green" title="E45">A98</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_6"><i><font color="Green" title="E48">A101</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_6"><i><font color="Green" title="E49">A102</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_6"><i><font color="Green" title="E51">A104</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_6"><i><font color="Green" title="E52">A105</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_6_9"><i><font color="Green" title="E59">A124</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_6_9"><i><font color="Green" title="E60">A125</font></i></a>, <a class="ref" href="gobrd13.html#T35" title="GOBRD13:th.35">GOBRD13:35</a></i>;<br/><a NAME="E6:33_1_1_6_9_1"/><b>then </b>
 <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>,0,<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>
 <b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_6_9_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="ref" href="jordan8.html#T21" title="JORDAN8:th.21">JORDAN8:21</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_6_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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_6"><i><font color="Green" title="E58">A111</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_6_10"/><i><font color="Green" title="E59">A126</font></i>: 
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E2:33_1_1_6_10"/><i><font color="Green" title="E60">A127</font></i>: 
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_6_10_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_6_10_1"/>
<font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 ;<br/><a NAME="E2:33_1_1_6_10_1"/><b>then </b>
( <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 &amp; <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:33_1_1_6_10_1"/><b>then </b>
<font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_6_10_1"/><b>then </b>
 <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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_6"><i><font color="Green" title="E45">A98</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_6"><i><font color="Green" title="E48">A101</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_6"><i><font color="Green" title="E49">A102</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_6"><i><font color="Green" title="E51">A104</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_6"><i><font color="Green" title="E52">A105</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_6_10"><i><font color="Green" title="E59">A126</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_6_10"><i><font color="Green" title="E60">A127</font></i></a>, <a class="ref" href="gobrd13.html#T37" title="GOBRD13:th.37">GOBRD13:37</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_6_10_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="ref" href="jordan8.html#T18" title="JORDAN8:th.18">JORDAN8:18</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_6_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_6"><i><font color="Green" title="E57">A110</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_6_11"/><i><font color="Green" title="E59">A128</font></i>: 
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E2:33_1_1_6_11"/><i><font color="Green" title="E60">A129</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_6_11_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_6_11_1"/>
<font color="Maroon" title="c14">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ;<br/><a NAME="E2:33_1_1_6_11_1"/><b>then </b>
<font color="Maroon" title="c14">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E3:33_1_1_6_11_1"/><b>then </b>
<font color="Maroon" title="c14">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54" title="BINARITH:th.54">BINARITH:54</a></i>;<br/><a NAME="E4:33_1_1_6_11_1"/><b>then </b>
<font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:33_1_1_6_11_1"/><b>then </b>
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_6"><i><font color="Green" title="E45">A98</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_6"><i><font color="Green" title="E48">A101</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_6"><i><font color="Green" title="E49">A102</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_6"><i><font color="Green" title="E51">A104</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_6"><i><font color="Green" title="E52">A105</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_6_11"><i><font color="Green" title="E59">A128</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_6_11"><i><font color="Green" title="E60">A129</font></i></a>, <a class="ref" href="gobrd13.html#T39" title="GOBRD13:th.39">GOBRD13:39</a></i>;<br/><a NAME="E6:33_1_1_6_11_1"/><b>then </b>
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,0 <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_6_11_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E14:33_1_1_6"><i><font color="Green" title="E58">A111</font></i></a>, <a class="ref" href="jordan8.html#T20" title="JORDAN8:th.20">JORDAN8:20</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_6_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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_6"><i><font color="Green" title="E58">A111</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_6_12"/><i><font color="Green" title="E59">A130</font></i>: 
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E2:33_1_1_6_12"/><i><font color="Green" title="E60">A131</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_6_12_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_6_12_1"/>
<font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 ;<br/><a NAME="E2:33_1_1_6_12_1"/><b>then </b>
( <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 &amp; <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:33_1_1_6_12_1"/><b>then </b>
<font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_6_12_1"/><b>then </b>
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_6"><i><font color="Green" title="E45">A98</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_6"><i><font color="Green" title="E48">A101</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_6"><i><font color="Green" title="E49">A102</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_6"><i><font color="Green" title="E51">A104</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_6"><i><font color="Green" title="E52">A105</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_6_12"><i><font color="Green" title="E59">A130</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_6_12"><i><font color="Green" title="E60">A131</font></i></a>, <a class="ref" href="gobrd13.html#T41" title="GOBRD13:th.41">GOBRD13:41</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_6_12_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_6"><i><font color="Green" title="E58">A111</font></i></a>, <a class="ref" href="jordan8.html#T19" title="JORDAN8:th.19">JORDAN8:19</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_6_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_6"><i><font color="Green" title="E57">A110</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/></div>
<b>end;</b></div></div><b>end;</b></div>
<a NAME="E18:33_1_1"/><i><font color="Green" title="E45">A132</font></i>: 
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#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><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> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_7"/><i><font color="Green" title="E46">A133</font></i>: 
<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>
 <b>and </b><br/><a NAME="E2:33_1_1_7"/><i><font color="Green" title="E47">A134</font></i>: 
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#K1" title="NAT_1:func.1">+</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> )
 ;<br/>

<a NAME="E3:33_1_1_7"/><i><font color="Green" title="E48">A135</font></i>: 
 <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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a></i>;<br/>
<a NAME="E4:33_1_1_7"/><i><font color="Green" title="E49">A136</font></i>: 
 <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a></i>;<br/>
<a NAME="E5:33_1_1_7"/><i><font color="Green" title="E50">A137</font></i>: 
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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E6:33_1_1_7"/><i><font color="Green" title="E51">A138</font></i>: 
2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T102" title="NEWTON:th.102">NEWTON:102</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_7_1"/>
( <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 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><b>by </b><i><a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_1"/><i><font color="Green" title="E52">A139</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>then consider </b><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> <b> such that </b><br/><a NAME="E3:33_1_1_7_1_1"/><i><font color="Green" title="E53">A140</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 <b>and </b><br/><a NAME="E4:33_1_1_7_1_1"/>
(  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 <b>and </b><br/><a NAME="E5:33_1_1_7_1_1"/><i><font color="Green" title="E54">A141</font></i>: 
<font 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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1"><i><font color="Green" title="E35">A80</font></i></a></i>;<br/><a NAME="E6:33_1_1_7_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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7_1_1"><i><font color="Green" title="E53">A140</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E7:33_1_1_7_1_1"/><b>then </b><i><font color="Green" title="E55">A142</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7"><i><font color="Green" title="E50">A137</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_1"><i><font color="Green" title="E53">A140</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/><div><i><font color="Green" title="E56">A143</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_1_1"><b>now </b></a><div class="add"><b>let </b><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> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E2:33_1_1_7_1_1_1"/><b>then </b>
( 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 )
 <b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_1"><i><font color="Green" title="E52">A139</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:33_1_1_7_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_7_1_1_1"/><b>then </b>
<span class="p1">(<span class="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="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>
 <b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1_7_1_1"><i><font color="Green" title="E54">A141</font></i></a>, <a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_7_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_1"><i><font color="Green" title="E55">A142</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_1_2"><b>now </b></a><div class="add"><b>let </b><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> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_1_2"/>
( <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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c12">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><a NAME="E2:33_1_1_7_1_1_2"/><b>then </b>
( 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 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c12">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_1"><i><font color="Green" title="E52">A139</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:33_1_1_7_1_1_2"/><b>then </b>
( <font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K3" title="REAL_1:func.3">+</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
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E8:33_1_1_7_1_1"><i><font color="Green" title="E56">A143</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/><b>let </b><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> ;<br/><b>assume </b><a NAME="E11:33_1_1_7_1_1"/><i><font color="Green" title="E57">A144</font></i>: 
( 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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><a NAME="E12:33_1_1_7_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#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E13:33_1_1_7_1_1"/><b>then </b>
<font color="Maroon" title="c12">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_1"><i><font color="Green" title="E52">A139</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_7_1_1"><i><font color="Green" title="E57">A144</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E14:33_1_1_7_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_7_1_1"><i><font color="Green" title="E57">A144</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_2"/><i><font color="Green" title="E52">A145</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>then consider </b><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> <b> such that </b><br/><a NAME="E3:33_1_1_7_1_2"/><i><font color="Green" title="E53">A146</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 <b>and </b><br/><a NAME="E4:33_1_1_7_1_2"/><i><font color="Green" title="E54">A147</font></i>: 
(  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 <b>and </b><br/><a NAME="E5:33_1_1_7_1_2"/><i><font color="Green" title="E55">A148</font></i>: 
<font 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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1"><i><font color="Green" title="E35">A80</font></i></a></i>;<br/><a NAME="E6:33_1_1_7_1_2"/><i><font color="Green" title="E56">A149</font></i>: 
( <span class="p1">(<span class="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1_7_1_2"><i><font color="Green" title="E55">A148</font></i></a>, <a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/><a NAME="E7:33_1_1_7_1_2"/><i><font color="Green" title="E57">A150</font></i>: 
( <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> &amp; 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#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7_1_2"><i><font color="Green" title="E53">A146</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E8:33_1_1_7_1_2"/><b>then </b><i><font color="Green" title="E58">A151</font></i>: 
( <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> &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="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7"><i><font color="Green" title="E50">A137</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_2"><i><font color="Green" title="E53">A146</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/><div><i><font color="Green" title="E59">A152</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_2_1"><b>now </b></a><div class="add"><b>let </b><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> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_2_1"/><i><font color="Green" title="E60">A153</font></i>: 
<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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E2:33_1_1_7_1_2_1"/><b>then </b>
( 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> 2 )
 <b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_2"><i><font color="Green" title="E52">A145</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:33_1_1_7_1_2_1"/><b>then </b>
( <font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 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 )
 <b>by </b><i><a class="ref" href="nat_1.html#T27" title="NAT_1:th.27">NAT_1:27</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_1_2_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_2"><i><font color="Green" title="E56">A149</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_7_1_2"><i><font color="Green" title="E58">A151</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_2_1"><i><font color="Green" title="E60">A153</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_2_2"><b>now </b></a><div class="add"><b>let </b><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> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_2_2"/><i><font color="Green" title="E60">A154</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c12">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><a NAME="E2:33_1_1_7_1_2_2"/><b>then </b>
( 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> 2 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c12">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 )
 <b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_2"><i><font color="Green" title="E52">A145</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:33_1_1_7_1_2_2"/><b>then </b><i><font color="Green" title="E61">A155</font></i>: 
( <font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 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 )
 <b>by </b><i><a class="ref" href="nat_1.html#T27" title="NAT_1:th.27">NAT_1:27</a></i>;<br/><b>let </b><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> ;<br/><b>assume </b><a NAME="E4:33_1_1_7_1_2_2"/><i><font color="Green" title="E62">A156</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><a NAME="E5:33_1_1_7_1_2_2"/>
( <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="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="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#K1" title="NAT_1:func.1">+</a> 1 &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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_2"><i><font color="Green" title="E52">A145</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_2"><i><font color="Green" title="E56">A149</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_7_1_2"><i><font color="Green" title="E58">A151</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_2_2"><i><font color="Green" title="E60">A154</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_2_2"><i><font color="Green" title="E61">A155</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_2_2"><i><font color="Green" title="E62">A156</font></i></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></i>;<br/><a NAME="E6:33_1_1_7_1_2_2"/><b>then </b>
(  <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 &amp;  <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> 0 )
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_7_1_2_2"/>
<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#K3" title="REAL_1:func.3">+</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
 <b>by </b><i><a class="ref" href="uniform1.html#T13" title="UNIFORM1:th.13">UNIFORM1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E11:33_1_1_7_1_2"/><i><font color="Green" title="E60">A157</font></i>: 
<font 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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_7_1_2"><i><font color="Green" title="E59">A152</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/><b>let </b><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> ;<br/><b>assume </b><a NAME="E12:33_1_1_7_1_2"/><i><font color="Green" title="E61">A158</font></i>: 
( 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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><a NAME="E13:33_1_1_7_1_2"/><b>then </b>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E14:33_1_1_7_1_2"/><b>then </b><i><font color="Green" title="E62">A159</font></i>: 
<font color="Maroon" title="c12">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_2"><i><font color="Green" title="E52">A145</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_7_1_2"><i><font color="Green" title="E61">A158</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E15:33_1_1_7_1_2"/><i><font color="Green" title="E63">A160</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E16:33_1_1_7_1_2"/><b>then </b><i><font color="Green" title="E64">A161</font></i>: 
 <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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_2"><i><font color="Green" title="E56">A149</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_7_1_2"><i><font color="Green" title="E58">A151</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_7_1_2"><i><font color="Green" title="E60">A157</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_7_1_2"><i><font color="Green" title="E61">A158</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_7_1_2"><i><font color="Green" title="E62">A159</font></i></a>, <a class="ref" href="gobrd13.html#D3" title="GOBRD13:def.3">GOBRD13:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 ;<br/><b>then consider </b><font color="Maroon" title="c13">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:33_1_1_7_1_2_3"/><i><font color="Green" title="E65">A162</font></i>: 
( <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> &amp; <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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E16:33_1_1_7_1_2"><i><font color="Green" title="E64">A161</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><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> <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7_1_2_3"><i><font color="Green" title="E65">A162</font></i></a></i>;<br/><b>reconsider </b><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="E6:33_1_1_7_1_2_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="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#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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#K21" title="EUCLID:func.21">`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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">s</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_2"><i><font color="Green" title="E53">A146</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_2"><i><font color="Green" title="E57">A150</font></i></a>, <a class="ref" href="gobrd11.html#T31" title="GOBRD11:th.31">GOBRD11:31</a></i>;<br/><b>then consider </b><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><b> such that </b><br/><a NAME="E8:33_1_1_7_1_2_3"/><i><font color="Green" title="E66">A163</font></i>: 
<font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c16">r</font>,<font color="Maroon" title="c17">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>and </b><br/><a NAME="E9:33_1_1_7_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#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">r</font> &amp; <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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>and </b><br/><a NAME="E10:33_1_1_7_1_2_3"/><i><font color="Green" title="E67">A164</font></i>: 
<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#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">s</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7_1_2_3"><i><font color="Green" title="E65">A162</font></i></a></i>;<br/><a NAME="E11:33_1_1_7_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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7"><i><font color="Green" title="E50">A137</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/><a NAME="E12:33_1_1_7_1_2_3"/><b>then </b><i><font color="Green" title="E68">A165</font></i>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="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#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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#K17" title="PSCOMP_1:func.17">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</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#K5" title="REAL_1:func.5">-</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#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="ref" href="jordan8.html#D1" title="JORDAN8:def.1">JORDAN8:def 1</a></i>;<br/><i><font color="Green" title="E69">A166</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</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#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</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>
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1_7"><i><font color="Green" title="E51">A138</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>
;<br/><a NAME="E14:33_1_1_7_1_2_3"/><i><font color="Green" title="E70">A167</font></i>: 
<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#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E12:33_1_1_7_1_2_3"><i><font color="Green" title="E68">A165</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><a NAME="E15:33_1_1_7_1_2_3"/>
 <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">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#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="jordan8.html#T12" title="JORDAN8:th.12">JORDAN8:12</a></i>;<br/><a NAME="E16:33_1_1_7_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">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#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E17:33_1_1_7_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1_7"><i><font color="Green" title="E51">A138</font></i></a>, <a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/><a NAME="E18:33_1_1_7_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E19:33_1_1_7_1_2_3"/><b>then </b><i><font color="Green" title="E71">A168</font></i>: 
 <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">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>
 <b>by </b><i><a class="txt" href="jordan9.html#E10:33_1_1_7_1_2_3"><i><font color="Green" title="E67">A164</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_7_1_2_3"><i><font color="Green" title="E69">A166</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_7_1_2_3"><i><font color="Green" title="E70">A167</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E20:33_1_1_7_1_2_3"/>
<font color="Maroon" title="c15">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7_1_2_3"><i><font color="Green" title="E65">A162</font></i></a>, <a class="ref" href="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></i>;<br/><b>hence </b><a NAME="E21:33_1_1_7_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E8:33_1_1_7_1_2_3"><i><font color="Green" title="E66">A163</font></i></a>, <a class="txt" href="jordan9.html#E19:33_1_1_7_1_2_3"><i><font color="Green" title="E71">A168</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E18:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 ;<br/><a NAME="E19:33_1_1_7_1_2"/><i><font color="Green" title="E65">A169</font></i>: 
 <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">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>
 <b>by </b><i><a class="ref" href="sprect_1.html#T13" title="SPRECT_1:th.13">SPRECT_1:13</a></i>;<br/><a NAME="E20:33_1_1_7_1_2"/>
 <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_2"><i><font color="Green" title="E54">A147</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_2"><i><font color="Green" title="E56">A149</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_7_1_2"><i><font color="Green" title="E58">A151</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_7_1_2"><i><font color="Green" title="E60">A157</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_7_1_2"><i><font color="Green" title="E61">A158</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_7_1_2"><i><font color="Green" title="E62">A159</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_7_1_2"><i><font color="Green" title="E63">A160</font></i></a>, <a class="ref" href="gobrd13.html#D2" title="GOBRD13:def.2">GOBRD13:def 2</a></i>;<br/><b>hence </b><a NAME="E21:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1_7_1_2"><i><font color="Green" title="E65">A169</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3"/><i><font color="Green" title="E52">A170</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:33_1_1_7_1_3"/><b>then </b><i><font color="Green" title="E53">A171</font></i>: 
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> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/><a NAME="E3:33_1_1_7_1_3"/><i><font color="Green" title="E54">A172</font></i>: 
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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3"><i><font color="Green" title="E52">A170</font></i></a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a></i>;<br/><a NAME="E4:33_1_1_7_1_3"/><i><font color="Green" title="E55">A173</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3"><i><font color="Green" title="E52">A170</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><b>then consider </b><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> <b> such that </b><br/><a NAME="E6:33_1_1_7_1_3"/><i><font color="Green" title="E56">A174</font></i>: 
( <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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>and </b><br/><a NAME="E7:33_1_1_7_1_3"/><i><font color="Green" title="E57">A175</font></i>: 
( <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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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> )
 <b>and </b><br/><a NAME="E8:33_1_1_7_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E9:33_1_1_7_1_3"/><i><font color="Green" title="E58">A176</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">i2</font> &amp; <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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j2</font> &amp; <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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E10:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="ref" href="binarith.html#T52" title="BINARITH:th.52">BINARITH:52</a></i>;<br/><a NAME="E11:33_1_1_7_1_3"/><b>then </b><i><font color="Green" title="E59">A177</font></i>: 
( <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="binarith.html#K3" title="BINARITH:func.3">-'</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> &amp;  <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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3"><i><font color="Green" title="E52">A170</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E12:33_1_1_7_1_3"/><i><font color="Green" title="E60">A178</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a></i>;<br/><a NAME="E13:33_1_1_7_1_3"/><i><font color="Green" title="E61">A179</font></i>: 
(  <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_7"><i><font color="Green" title="E47">A134</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a></i>;<br/><a NAME="E14:33_1_1_7_1_3"/><i><font color="Green" title="E62">A180</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E17:33_1_1"><i><font color="Green" title="E44">A97</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7"><i><font color="Green" title="E47">A134</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3"><i><font color="Green" title="E52">A170</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a></i>;<br/><a NAME="E15:33_1_1_7_1_3"/><i><font color="Green" title="E63">A181</font></i>: 
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E17:33_1_1"><i><font color="Green" title="E44">A97</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7"><i><font color="Green" title="E47">A134</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3"><i><font color="Green" title="E52">A170</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a></i>;<br/><a NAME="E16:33_1_1_7_1_3"/><i><font color="Green" title="E64">A182</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E17:33_1_1"><i><font color="Green" title="E44">A97</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7"><i><font color="Green" title="E47">A134</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3"><i><font color="Green" title="E52">A170</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a></i>;<br/><a NAME="E17:33_1_1_7_1_3"/><i><font color="Green" title="E65">A183</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E17:33_1_1"><i><font color="Green" title="E44">A97</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7"><i><font color="Green" title="E47">A134</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3"><i><font color="Green" title="E52">A170</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a></i>;<br/><a NAME="E18:33_1_1_7_1_3"/><i><font color="Green" title="E66">A184</font></i>: 
(  <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E17:33_1_1"><i><font color="Green" title="E44">A97</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7"><i><font color="Green" title="E47">A134</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3"><i><font color="Green" title="E52">A170</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a></i>;<br/><a NAME="E19:33_1_1_7_1_3"/><i><font color="Green" title="E67">A185</font></i>: 
(  <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E17:33_1_1"><i><font color="Green" title="E44">A97</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7"><i><font color="Green" title="E47">A134</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3"><i><font color="Green" title="E52">A170</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a></i>;<br/><a NAME="E20:33_1_1_7_1_3"/><i><font color="Green" title="E68">A186</font></i>: 
(  <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E17:33_1_1"><i><font color="Green" title="E44">A97</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7"><i><font color="Green" title="E47">A134</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3"><i><font color="Green" title="E52">A170</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a></i>;<br/><a NAME="E21:33_1_1_7_1_3"/><i><font color="Green" title="E69">A187</font></i>: 
(  <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E17:33_1_1"><i><font color="Green" title="E44">A97</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7"><i><font color="Green" title="E47">A134</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3"><i><font color="Green" title="E52">A170</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a></i>;<br/><a NAME="E22:33_1_1_7_1_3"/><i><font color="Green" title="E70">A188</font></i>: 
(  <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E17:33_1_1"><i><font color="Green" title="E44">A97</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7"><i><font color="Green" title="E47">A134</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3"><i><font color="Green" title="E52">A170</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a></i>;<br/><a NAME="E23:33_1_1_7_1_3"/><i><font color="Green" title="E71">A189</font></i>: 
(  <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E17:33_1_1"><i><font color="Green" title="E44">A97</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7"><i><font color="Green" title="E47">A134</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3"><i><font color="Green" title="E52">A170</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a></i>;<br/><a NAME="E24:33_1_1_7_1_3"/><i><font color="Green" title="E72">A190</font></i>: 
(  <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E17:33_1_1"><i><font color="Green" title="E44">A97</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7"><i><font color="Green" title="E47">A134</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3"><i><font color="Green" title="E52">A170</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a></i>;<br/><a NAME="E25:33_1_1_7_1_3"/><i><font color="Green" title="E73">A191</font></i>: 
(  <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E17:33_1_1"><i><font color="Green" title="E44">A97</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7"><i><font color="Green" title="E47">A134</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3"><i><font color="Green" title="E52">A170</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a></i>;<br/><b>thus </b><a NAME="E26:33_1_1_7_1_3"/><i><font color="Green" title="E74">A192</font></i>: 
<font 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#K1" title="NAT_1:func.1">+</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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="33_1_1_7_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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> )
 ;<br/><div class="add"><b>then consider </b><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> <b> such that </b><br/><a NAME="E3:33_1_1_7_1_3_1_1_1"/><i><font color="Green" title="E75">A193</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>and </b><br/><a NAME="E4:33_1_1_7_1_3_1_1_1"/><i><font color="Green" title="E76">A194</font></i>: 
<font 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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1"><i><font color="Green" title="E35">A80</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3"><i><font color="Green" title="E52">A170</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_7_1_3"><i><font color="Green" title="E61">A179</font></i></a></i>;<br/><b>thus </b><a NAME="E5:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_1_1"><b>proof </b></a><div class="add">

<b>set </b><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="E1:33_1_1_7_1_3_1_1_1_1"/><i><font color="Green" title="E77">A195</font></i>: 
( <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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_7_1_3"><i><font color="Green" title="E59">A177</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E2:33_1_1_7_1_3_1_1_1_1"/><i><font color="Green" title="E78">A196</font></i>: 
<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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_7_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#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_7_1_3"><i><font color="Green" title="E60">A178</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3_1_1_1"><i><font color="Green" title="E75">A193</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_1_1"><i><font color="Green" title="E77">A195</font></i></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_1_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E79">A197</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E80">A198</font></i>: 
<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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">i1'</font>, <font color="Maroon" title="c19">j1'</font>, <font color="Maroon" title="c20">i2'</font>, <font color="Maroon" title="c21">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> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_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"><font color="Maroon" title="c18">i1'</font>,<font color="Maroon" title="c19">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="c20">i2'</font>,<font color="Maroon" title="c21">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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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">i1'</font>,<font color="Maroon" title="c19">j1'</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#K1" title="NAT_1:func.1">+</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">i2'</font>,<font color="Maroon" title="c21">j2'</font> )
 ;<br/><a NAME="E2:33_1_1_7_1_3_1_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1'</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1'</font> &amp; <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2'</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_7_1_3"><i><font color="Green" title="E62">A180</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_1_1_1_1"><i><font color="Green" title="E79">A197</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_1_1_1_1_1_1_1"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_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">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3"><i><font color="Green" title="E53">A171</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_7_1_3"><i><font color="Green" title="E62">A180</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3_1_1_1"><i><font color="Green" title="E76">A194</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_1_1"><i><font color="Green" title="E78">A196</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_1_1_1_1"><i><font color="Green" title="E79">A197</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_1_1_1_1"><i><font color="Green" title="E80">A198</font></i></a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_1_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_1_1_1_1_1_2"/><i><font color="Green" title="E79">A199</font></i>: 
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_1_1_1_1_1_2"/><i><font color="Green" title="E80">A200</font></i>: 
<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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">i1'</font>, <font color="Maroon" title="c19">j1'</font>, <font color="Maroon" title="c20">i2'</font>, <font color="Maroon" title="c21">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> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_3_1_1_1_1_1_2_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i1'</font>,<font color="Maroon" title="c19">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="c20">i2'</font>,<font color="Maroon" title="c21">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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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">i1'</font>,<font color="Maroon" title="c19">j1'</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="binarith.html#K3" title="BINARITH:func.3">-'</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">i2'</font>,<font color="Maroon" title="c21">j2'</font> )
 ;<br/><a NAME="E2:33_1_1_7_1_3_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E81">A201</font></i>: 
( <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1'</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1'</font> &amp; <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2'</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_7_1_3"><i><font color="Green" title="E63">A181</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_1_1_1_2"><i><font color="Green" title="E79">A199</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_1_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c19">j1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">j2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_7_1_3"><i><font color="Green" title="E58">A176</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E4:33_1_1_7_1_3_1_1_1_1_1_2_1"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">j2'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_1_1_1_2_1"><i><font color="Green" title="E81">A201</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_7_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">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="uniform1.html#T13" title="UNIFORM1:th.13">UNIFORM1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3"><i><font color="Green" title="E53">A171</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_7_1_3"><i><font color="Green" title="E63">A181</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3_1_1_1"><i><font color="Green" title="E76">A194</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_1_1"><i><font color="Green" title="E78">A196</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_1_1_1_2"><i><font color="Green" title="E79">A199</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_1_1_1_2"><i><font color="Green" title="E80">A200</font></i></a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_1_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_1_1_1_1_1_3"/><i><font color="Green" title="E79">A202</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_1_1_1_1_1_3"/><i><font color="Green" title="E80">A203</font></i>: 
<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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_1_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">i1'</font>, <font color="Maroon" title="c19">j1'</font>, <font color="Maroon" title="c20">i2'</font>, <font color="Maroon" title="c21">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> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_3_1_1_1_1_1_3_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i1'</font>,<font color="Maroon" title="c19">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="c20">i2'</font>,<font color="Maroon" title="c21">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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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">i1'</font>,<font color="Maroon" title="c19">j1'</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#K1" title="NAT_1:func.1">+</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">i2'</font>,<font color="Maroon" title="c21">j2'</font> )
 ;<br/><a NAME="E2:33_1_1_7_1_3_1_1_1_1_1_3_1"/><b>then </b>
( <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1'</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1'</font> &amp; <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2'</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E16:33_1_1_7_1_3"><i><font color="Green" title="E64">A182</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_1_1_1_3"><i><font color="Green" title="E79">A202</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_1_1_1_1_1_3_1"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_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">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3"><i><font color="Green" title="E53">A171</font></i></a>, <a class="txt" href="jordan9.html#E16:33_1_1_7_1_3"><i><font color="Green" title="E64">A182</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3_1_1_1"><i><font color="Green" title="E76">A194</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_1_1"><i><font color="Green" title="E78">A196</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_1_1_1_3"><i><font color="Green" title="E79">A202</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_1_1_1_3"><i><font color="Green" title="E80">A203</font></i></a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_1_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_1_1_1_1_1_4"/><i><font color="Green" title="E79">A204</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_1_1_1_1_1_4"/><i><font color="Green" title="E80">A205</font></i>: 
<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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_1_1_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">i1'</font>, <font color="Maroon" title="c19">j1'</font>, <font color="Maroon" title="c20">i2'</font>, <font color="Maroon" title="c21">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> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_3_1_1_1_1_1_4_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i1'</font>,<font color="Maroon" title="c19">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="c20">i2'</font>,<font color="Maroon" title="c21">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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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">i1'</font>,<font color="Maroon" title="c19">j1'</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="binarith.html#K3" title="BINARITH:func.3">-'</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">i2'</font>,<font color="Maroon" title="c21">j2'</font> )
 ;<br/><a NAME="E2:33_1_1_7_1_3_1_1_1_1_1_4_1"/><b>then </b><i><font color="Green" title="E81">A206</font></i>: 
( <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1'</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1'</font> &amp; <font color="Maroon" title="c13">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2'</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E17:33_1_1_7_1_3"><i><font color="Green" title="E65">A183</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_1_1_1_4"><i><font color="Green" title="E79">A204</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_1_1_1_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c18">i1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">i2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_7_1_3"><i><font color="Green" title="E58">A176</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E4:33_1_1_7_1_3_1_1_1_1_1_4_1"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">i2'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_1_1_1_4_1"><i><font color="Green" title="E81">A206</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_7_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">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="uniform1.html#T13" title="UNIFORM1:th.13">UNIFORM1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3"><i><font color="Green" title="E53">A171</font></i></a>, <a class="txt" href="jordan9.html#E17:33_1_1_7_1_3"><i><font color="Green" title="E65">A183</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3_1_1_1"><i><font color="Green" title="E76">A194</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_1_1"><i><font color="Green" title="E78">A196</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_1_1_1_4"><i><font color="Green" title="E79">A204</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_1_1_1_4"><i><font color="Green" title="E80">A205</font></i></a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3_1_1_2"/><i><font color="Green" title="E75">A207</font></i>: 
(  <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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/><div class="add"><b>then consider </b><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> <b> such that </b><br/><a NAME="E3:33_1_1_7_1_3_1_1_2"/><i><font color="Green" title="E76">A208</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>and </b><br/><a NAME="E4:33_1_1_7_1_3_1_1_2"/><i><font color="Green" title="E77">A209</font></i>: 
<font 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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1"><i><font color="Green" title="E35">A80</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3"><i><font color="Green" title="E52">A170</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_7_1_3"><i><font color="Green" title="E61">A179</font></i></a></i>;<br/><b>thus </b><a NAME="E5:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_2_1"><b>proof </b></a><div class="add">

<b>set </b><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="E1:33_1_1_7_1_3_1_1_2_1"/><i><font color="Green" title="E78">A210</font></i>: 
( <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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_7_1_3"><i><font color="Green" title="E59">A177</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E2:33_1_1_7_1_3_1_1_2_1"/><i><font color="Green" title="E79">A211</font></i>: 
<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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_7_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#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_7_1_3"><i><font color="Green" title="E60">A178</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3_1_1_2"><i><font color="Green" title="E76">A208</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_2_1"><i><font color="Green" title="E78">A210</font></i></a>, <a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_2_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_1_1_2_1_1_1"/><i><font color="Green" title="E80">A212</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_1_1_2_1_1_1"/><i><font color="Green" title="E81">A213</font></i>: 
<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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">i1'</font>, <font color="Maroon" title="c19">j1'</font>, <font color="Maroon" title="c20">i2'</font>, <font color="Maroon" title="c21">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> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_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"><font color="Maroon" title="c18">i1'</font>,<font color="Maroon" title="c19">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="c20">i2'</font>,<font color="Maroon" title="c21">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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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">i1'</font>,<font color="Maroon" title="c19">j1'</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#K1" title="NAT_1:func.1">+</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">i2'</font>,<font color="Maroon" title="c21">j2'</font> )
 ;<br/><a NAME="E2:33_1_1_7_1_3_1_1_2_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1'</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1'</font> &amp; <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2'</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_7_1_3"><i><font color="Green" title="E66">A184</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_2"><i><font color="Green" title="E75">A207</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_2_1_1_1"><i><font color="Green" title="E80">A212</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_1_1_2_1_1_1_1"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_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">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3"><i><font color="Green" title="E53">A171</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_7_1_3"><i><font color="Green" title="E66">A184</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_2"><i><font color="Green" title="E75">A207</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3_1_1_2"><i><font color="Green" title="E77">A209</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_2_1"><i><font color="Green" title="E79">A211</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_2_1_1_1"><i><font color="Green" title="E80">A212</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_2_1_1_1"><i><font color="Green" title="E81">A213</font></i></a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_2_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_1_1_2_1_1_2"/><i><font color="Green" title="E80">A214</font></i>: 
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_1_1_2_1_1_2"/><i><font color="Green" title="E81">A215</font></i>: 
<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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">i1'</font>, <font color="Maroon" title="c19">j1'</font>, <font color="Maroon" title="c20">i2'</font>, <font color="Maroon" title="c21">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> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_3_1_1_2_1_1_2_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i1'</font>,<font color="Maroon" title="c19">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="c20">i2'</font>,<font color="Maroon" title="c21">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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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">i1'</font>,<font color="Maroon" title="c19">j1'</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#K1" title="NAT_1:func.1">+</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">i2'</font>,<font color="Maroon" title="c21">j2'</font> )
 ;<br/><a NAME="E2:33_1_1_7_1_3_1_1_2_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1'</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1'</font> &amp; <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2'</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E19:33_1_1_7_1_3"><i><font color="Green" title="E67">A185</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_2"><i><font color="Green" title="E75">A207</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_2_1_1_2"><i><font color="Green" title="E80">A214</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_1_1_2_1_1_2_1"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_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">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3"><i><font color="Green" title="E53">A171</font></i></a>, <a class="txt" href="jordan9.html#E19:33_1_1_7_1_3"><i><font color="Green" title="E67">A185</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_2"><i><font color="Green" title="E75">A207</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3_1_1_2"><i><font color="Green" title="E77">A209</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_2_1"><i><font color="Green" title="E79">A211</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_2_1_1_2"><i><font color="Green" title="E80">A214</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_2_1_1_2"><i><font color="Green" title="E81">A215</font></i></a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_2_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_1_1_2_1_1_3"/><i><font color="Green" title="E80">A216</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_1_1_2_1_1_3"/><i><font color="Green" title="E81">A217</font></i>: 
<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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_2_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">i1'</font>, <font color="Maroon" title="c19">j1'</font>, <font color="Maroon" title="c20">i2'</font>, <font color="Maroon" title="c21">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> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_3_1_1_2_1_1_3_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i1'</font>,<font color="Maroon" title="c19">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="c20">i2'</font>,<font color="Maroon" title="c21">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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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">i1'</font>,<font color="Maroon" title="c19">j1'</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="binarith.html#K3" title="BINARITH:func.3">-'</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">i2'</font>,<font color="Maroon" title="c21">j2'</font> )
 ;<br/><a NAME="E2:33_1_1_7_1_3_1_1_2_1_1_3_1"/><b>then </b><i><font color="Green" title="E82">A218</font></i>: 
( <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1'</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1'</font> &amp; <font color="Maroon" title="c13">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2'</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_7_1_3"><i><font color="Green" title="E68">A186</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_2"><i><font color="Green" title="E75">A207</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_2_1_1_3"><i><font color="Green" title="E80">A216</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_1_1_2_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c18">i1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">i2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_7_1_3"><i><font color="Green" title="E58">A176</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E4:33_1_1_7_1_3_1_1_2_1_1_3_1"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">i2'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_2_1_1_3_1"><i><font color="Green" title="E82">A218</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_7_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">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="uniform1.html#T13" title="UNIFORM1:th.13">UNIFORM1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3"><i><font color="Green" title="E53">A171</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_7_1_3"><i><font color="Green" title="E68">A186</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_2"><i><font color="Green" title="E75">A207</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3_1_1_2"><i><font color="Green" title="E77">A209</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_2_1"><i><font color="Green" title="E79">A211</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_2_1_1_3"><i><font color="Green" title="E80">A216</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_2_1_1_3"><i><font color="Green" title="E81">A217</font></i></a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_2_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_1_1_2_1_1_4"/><i><font color="Green" title="E80">A219</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_1_1_2_1_1_4"/><i><font color="Green" title="E81">A220</font></i>: 
<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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_2_1_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">i1'</font>, <font color="Maroon" title="c19">j1'</font>, <font color="Maroon" title="c20">i2'</font>, <font color="Maroon" title="c21">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> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_3_1_1_2_1_1_4_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i1'</font>,<font color="Maroon" title="c19">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="c20">i2'</font>,<font color="Maroon" title="c21">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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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">i1'</font>,<font color="Maroon" title="c19">j1'</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="binarith.html#K3" title="BINARITH:func.3">-'</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">i2'</font>,<font color="Maroon" title="c21">j2'</font> )
 ;<br/><a NAME="E2:33_1_1_7_1_3_1_1_2_1_1_4_1"/><b>then </b><i><font color="Green" title="E82">A221</font></i>: 
( <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1'</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1'</font> &amp; <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2'</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E21:33_1_1_7_1_3"><i><font color="Green" title="E69">A187</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_2"><i><font color="Green" title="E75">A207</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_2_1_1_4"><i><font color="Green" title="E80">A219</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_1_1_2_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c19">j1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">j2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_7_1_3"><i><font color="Green" title="E58">A176</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E4:33_1_1_7_1_3_1_1_2_1_1_4_1"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">j2'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_2_1_1_4_1"><i><font color="Green" title="E82">A221</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_7_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">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="uniform1.html#T13" title="UNIFORM1:th.13">UNIFORM1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3"><i><font color="Green" title="E53">A171</font></i></a>, <a class="txt" href="jordan9.html#E21:33_1_1_7_1_3"><i><font color="Green" title="E69">A187</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_2"><i><font color="Green" title="E75">A207</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3_1_1_2"><i><font color="Green" title="E77">A209</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_2_1"><i><font color="Green" title="E79">A211</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_2_1_1_4"><i><font color="Green" title="E80">A219</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_2_1_1_4"><i><font color="Green" title="E81">A220</font></i></a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3_1_1_3"/><i><font color="Green" title="E75">A222</font></i>: 
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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/><div class="add"><b>then consider </b><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> <b> such that </b><br/><a NAME="E3:33_1_1_7_1_3_1_1_3"/><i><font color="Green" title="E76">A223</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>and </b><br/><a NAME="E4:33_1_1_7_1_3_1_1_3"/><i><font color="Green" title="E77">A224</font></i>: 
<font 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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1"><i><font color="Green" title="E35">A80</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3"><i><font color="Green" title="E52">A170</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_7_1_3"><i><font color="Green" title="E61">A179</font></i></a></i>;<br/><b>thus </b><a NAME="E5:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_3_1"><b>proof </b></a><div class="add">

<b>set </b><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="E1:33_1_1_7_1_3_1_1_3_1"/><i><font color="Green" title="E78">A225</font></i>: 
( <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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_7_1_3"><i><font color="Green" title="E59">A177</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E2:33_1_1_7_1_3_1_1_3_1"/><i><font color="Green" title="E79">A226</font></i>: 
<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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_7_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#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_7_1_3"><i><font color="Green" title="E60">A178</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3_1_1_3"><i><font color="Green" title="E76">A223</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_3_1"><i><font color="Green" title="E78">A225</font></i></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_3_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_1_1_3_1_1_1"/><i><font color="Green" title="E80">A227</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_1_1_3_1_1_1"/><i><font color="Green" title="E81">A228</font></i>: 
<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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_3_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">i1'</font>, <font color="Maroon" title="c19">j1'</font>, <font color="Maroon" title="c20">i2'</font>, <font color="Maroon" title="c21">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> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_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"><font color="Maroon" title="c18">i1'</font>,<font color="Maroon" title="c19">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="c20">i2'</font>,<font color="Maroon" title="c21">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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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">i1'</font>,<font color="Maroon" title="c19">j1'</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="binarith.html#K3" title="BINARITH:func.3">-'</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">i2'</font>,<font color="Maroon" title="c21">j2'</font> )
 ;<br/><a NAME="E2:33_1_1_7_1_3_1_1_3_1_1_1_1"/><b>then </b><i><font color="Green" title="E82">A229</font></i>: 
( <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1'</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1'</font> &amp; <font color="Maroon" title="c13">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2'</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_7_1_3"><i><font color="Green" title="E70">A188</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_3"><i><font color="Green" title="E75">A222</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_3_1_1_1"><i><font color="Green" title="E80">A227</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_1_1_3_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c18">i1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">i2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_7_1_3"><i><font color="Green" title="E58">A176</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E4:33_1_1_7_1_3_1_1_3_1_1_1_1"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">i2'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_3_1_1_1_1"><i><font color="Green" title="E82">A229</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_7_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">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="uniform1.html#T13" title="UNIFORM1:th.13">UNIFORM1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3"><i><font color="Green" title="E53">A171</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_7_1_3"><i><font color="Green" title="E70">A188</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_3"><i><font color="Green" title="E75">A222</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3_1_1_3"><i><font color="Green" title="E77">A224</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_3_1"><i><font color="Green" title="E79">A226</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_3_1_1_1"><i><font color="Green" title="E80">A227</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_3_1_1_1"><i><font color="Green" title="E81">A228</font></i></a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_3_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_1_1_3_1_1_2"/><i><font color="Green" title="E80">A230</font></i>: 
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_1_1_3_1_1_2"/><i><font color="Green" title="E81">A231</font></i>: 
<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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_3_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">i1'</font>, <font color="Maroon" title="c19">j1'</font>, <font color="Maroon" title="c20">i2'</font>, <font color="Maroon" title="c21">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> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_3_1_1_3_1_1_2_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i1'</font>,<font color="Maroon" title="c19">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="c20">i2'</font>,<font color="Maroon" title="c21">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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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">i1'</font>,<font color="Maroon" title="c19">j1'</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#K1" title="NAT_1:func.1">+</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">i2'</font>,<font color="Maroon" title="c21">j2'</font> )
 ;<br/><a NAME="E2:33_1_1_7_1_3_1_1_3_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1'</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1'</font> &amp; <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2'</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_7_1_3"><i><font color="Green" title="E71">A189</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_3"><i><font color="Green" title="E75">A222</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_3_1_1_2"><i><font color="Green" title="E80">A230</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_1_1_3_1_1_2_1"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_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">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3"><i><font color="Green" title="E53">A171</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_7_1_3"><i><font color="Green" title="E71">A189</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_3"><i><font color="Green" title="E75">A222</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3_1_1_3"><i><font color="Green" title="E77">A224</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_3_1"><i><font color="Green" title="E79">A226</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_3_1_1_2"><i><font color="Green" title="E80">A230</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_3_1_1_2"><i><font color="Green" title="E81">A231</font></i></a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_3_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_1_1_3_1_1_3"/><i><font color="Green" title="E80">A232</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_1_1_3_1_1_3"/><i><font color="Green" title="E81">A233</font></i>: 
<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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_3_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">i1'</font>, <font color="Maroon" title="c19">j1'</font>, <font color="Maroon" title="c20">i2'</font>, <font color="Maroon" title="c21">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> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_3_1_1_3_1_1_3_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i1'</font>,<font color="Maroon" title="c19">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="c20">i2'</font>,<font color="Maroon" title="c21">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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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">i1'</font>,<font color="Maroon" title="c19">j1'</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="binarith.html#K3" title="BINARITH:func.3">-'</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">i2'</font>,<font color="Maroon" title="c21">j2'</font> )
 ;<br/><a NAME="E2:33_1_1_7_1_3_1_1_3_1_1_3_1"/><b>then </b><i><font color="Green" title="E82">A234</font></i>: 
( <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1'</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1'</font> &amp; <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2'</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_7_1_3"><i><font color="Green" title="E72">A190</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_3"><i><font color="Green" title="E75">A222</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_3_1_1_3"><i><font color="Green" title="E80">A232</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_1_1_3_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c19">j1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">j2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_7_1_3"><i><font color="Green" title="E58">A176</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E4:33_1_1_7_1_3_1_1_3_1_1_3_1"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">j2'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_3_1_1_3_1"><i><font color="Green" title="E82">A234</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_7_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">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="uniform1.html#T13" title="UNIFORM1:th.13">UNIFORM1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3"><i><font color="Green" title="E53">A171</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_7_1_3"><i><font color="Green" title="E72">A190</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_3"><i><font color="Green" title="E75">A222</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3_1_1_3"><i><font color="Green" title="E77">A224</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_3_1"><i><font color="Green" title="E79">A226</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_3_1_1_3"><i><font color="Green" title="E80">A232</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_3_1_1_3"><i><font color="Green" title="E81">A233</font></i></a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_3_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_1_1_3_1_1_4"/><i><font color="Green" title="E80">A235</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_1_1_3_1_1_4"/><i><font color="Green" title="E81">A236</font></i>: 
<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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_3_1_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">i1'</font>, <font color="Maroon" title="c19">j1'</font>, <font color="Maroon" title="c20">i2'</font>, <font color="Maroon" title="c21">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> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_3_1_1_3_1_1_4_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i1'</font>,<font color="Maroon" title="c19">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="c20">i2'</font>,<font color="Maroon" title="c21">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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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">i1'</font>,<font color="Maroon" title="c19">j1'</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#K1" title="NAT_1:func.1">+</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">i2'</font>,<font color="Maroon" title="c21">j2'</font> )
 ;<br/><a NAME="E2:33_1_1_7_1_3_1_1_3_1_1_4_1"/><b>then </b>
( <font color="Maroon" title="c13">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1'</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1'</font> &amp; <font color="Maroon" title="c13">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2'</font> &amp; <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E25:33_1_1_7_1_3"><i><font color="Green" title="E73">A191</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_3"><i><font color="Green" title="E75">A222</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_3_1_1_4"><i><font color="Green" title="E80">A235</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_1_1_3_1_1_4_1"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_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">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3"><i><font color="Green" title="E53">A171</font></i></a>, <a class="txt" href="jordan9.html#E25:33_1_1_7_1_3"><i><font color="Green" title="E73">A191</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_3"><i><font color="Green" title="E75">A222</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3_1_1_3"><i><font color="Green" title="E77">A224</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_3_1"><i><font color="Green" title="E79">A226</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_1_1_3_1_1_4"><i><font color="Green" title="E80">A235</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_1_1_3_1_1_4"><i><font color="Green" title="E81">A236</font></i></a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>let </b><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> ;<br/><b>assume </b><a NAME="E27:33_1_1_7_1_3"/><i><font color="Green" title="E75">A237</font></i>: 
( 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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_7_1_3_2_1"/>
( <font color="Maroon" title="c15">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> or <font color="Maroon" title="c15">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3_2_1_1"/><i><font color="Green" title="E76">A238</font></i>: 
<font color="Maroon" title="c15">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:33_1_1_7_1_3_2_1_1"/><i><font color="Green" title="E77">A239</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i2</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_7_1_3"><i><font color="Green" title="E58">A176</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_2_1_1"/><i><font color="Green" title="E78">A240</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_7_1_3"><i><font color="Green" title="E58">A176</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><b>thus </b><a NAME="E4:33_1_1_7_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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="33_1_1_7_1_3_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_1"/><i><font color="Green" title="E79">A241</font></i>: 
(  <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c16">i</font>, <font color="Maroon" title="c17">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> <b> such that </b><br/><a NAME="E3:33_1_1_7_1_3_2_1_1_1_1_1"/><i><font color="Green" title="E80">A242</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="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="c16">i</font>,<font color="Maroon" title="c17">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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>and </b><br/><a NAME="E4:33_1_1_7_1_3_2_1_1_1_1_1"/><i><font color="Green" title="E81">A243</font></i>: 
<font 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#K1" title="NAT_1:func.1">+</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="c16">i</font>,<font color="Maroon" title="c17">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1"><i><font color="Green" title="E35">A80</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3"><i><font color="Green" title="E52">A170</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_7_1_3"><i><font color="Green" title="E61">A179</font></i></a></i>;<br/><a NAME="E5:33_1_1_7_1_3_2_1_1_1_1_1"/><i><font color="Green" title="E82">A244</font></i>: 
( <span class="p1">(<span class="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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"><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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_7_1_3"><i><font color="Green" title="E59">A177</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3_2_1_1_1_1_1"><i><font color="Green" title="E81">A243</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_7_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_7_1_3"><i><font color="Green" title="E60">A178</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3_2_1_1_1_1_1"><i><font color="Green" title="E80">A242</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3_2_1_1_1_1_1"><i><font color="Green" title="E81">A243</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_1"><i><font color="Green" title="E82">A244</font></i></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_1_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E83">A245</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_2_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E84">A246</font></i>: 
<span class="p1">(<span class="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 ;<br/><div class="add"><a NAME="E3:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_1_1_1_1"><i><font color="Green" title="E83">A245</font></i></a>, <a class="ref" href="gobrd13.html#T36" title="GOBRD13:th.36">GOBRD13:36</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_7_1_3"><i><font color="Green" title="E62">A180</font></i></a>, <a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E76">A238</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_1"><i><font color="Green" title="E79">A241</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_1"><i><font color="Green" title="E82">A244</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_1_1_1_1"><i><font color="Green" title="E83">A245</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1_1_1_1_1_1_1"><i><font color="Green" title="E84">A246</font></i></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></i>;<br/><a NAME="E5:33_1_1_7_1_3_2_1_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c14">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_7_1_3"><i><font color="Green" title="E61">A179</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_1_1_1_1"><i><font color="Green" title="E83">A245</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_7_1_3"><i><font color="Green" title="E62">A180</font></i></a>, <a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E76">A238</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_1"><i><font color="Green" title="E82">A244</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_1_1_1_1"><i><font color="Green" title="E83">A245</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1_1_1_1_1_1_1"><i><font color="Green" title="E84">A246</font></i></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_1_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E83">A247</font></i>: 
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_2_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E84">A248</font></i>: 
<span class="p1">(<span class="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E3:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_1_1_1_2"><i><font color="Green" title="E83">A247</font></i></a>, <a class="ref" href="gobrd13.html#T38" title="GOBRD13:th.38">GOBRD13:38</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_7_1_3"><i><font color="Green" title="E63">A181</font></i></a>, <a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E76">A238</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E78">A240</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_1"><i><font color="Green" title="E79">A241</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_1"><i><font color="Green" title="E82">A244</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_1_1_1_2"><i><font color="Green" title="E83">A247</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1_1_1_1_1_1_2"><i><font color="Green" title="E84">A248</font></i></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></i>;<br/><a NAME="E5:33_1_1_7_1_3_2_1_1_1_1_1_1_1_2"/>
( <font color="Maroon" title="c13">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_7_1_3"><i><font color="Green" title="E61">A179</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_1_1_1_2"><i><font color="Green" title="E83">A247</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_7_1_3"><i><font color="Green" title="E63">A181</font></i></a>, <a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E76">A238</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E78">A240</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_1"><i><font color="Green" title="E82">A244</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_1_1_1_2"><i><font color="Green" title="E83">A247</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1_1_1_1_1_1_2"><i><font color="Green" title="E84">A248</font></i></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_1_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_1_1_1_3"/><i><font color="Green" title="E83">A249</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_2_1_1_1_1_1_1_1_3"/><i><font color="Green" title="E84">A250</font></i>: 
<span class="p1">(<span class="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E3:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_1_1_1_3"><i><font color="Green" title="E83">A249</font></i></a>, <a class="ref" href="gobrd13.html#T40" title="GOBRD13:th.40">GOBRD13:40</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E16:33_1_1_7_1_3"><i><font color="Green" title="E64">A182</font></i></a>, <a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E76">A238</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_1"><i><font color="Green" title="E79">A241</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_1"><i><font color="Green" title="E82">A244</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_1_1_1_3"><i><font color="Green" title="E83">A249</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1_1_1_1_1_1_3"><i><font color="Green" title="E84">A250</font></i></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></i>;<br/><a NAME="E5:33_1_1_7_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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_7_1_3"><i><font color="Green" title="E61">A179</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_1_1_1_3"><i><font color="Green" title="E83">A249</font></i></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E16:33_1_1_7_1_3"><i><font color="Green" title="E64">A182</font></i></a>, <a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E76">A238</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_1"><i><font color="Green" title="E82">A244</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_1_1_1_3"><i><font color="Green" title="E83">A249</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1_1_1_1_1_1_3"><i><font color="Green" title="E84">A250</font></i></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_1_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_1_1_1_4"/><i><font color="Green" title="E83">A251</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_2_1_1_1_1_1_1_1_4"/><i><font color="Green" title="E84">A252</font></i>: 
<span class="p1">(<span class="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 ;<br/><div class="add"><a NAME="E3:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_1_1_1_4"><i><font color="Green" title="E83">A251</font></i></a>, <a class="ref" href="gobrd13.html#T42" title="GOBRD13:th.42">GOBRD13:42</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E17:33_1_1_7_1_3"><i><font color="Green" title="E65">A183</font></i></a>, <a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E76">A238</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E77">A239</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_1"><i><font color="Green" title="E79">A241</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_1"><i><font color="Green" title="E82">A244</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_1_1_1_4"><i><font color="Green" title="E83">A251</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1_1_1_1_1_1_4"><i><font color="Green" title="E84">A252</font></i></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></i>;<br/><a NAME="E5:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_7_1_3"><i><font color="Green" title="E61">A179</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_1_1_1_4"><i><font color="Green" title="E83">A251</font></i></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E17:33_1_1_7_1_3"><i><font color="Green" title="E65">A183</font></i></a>, <a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E76">A238</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E77">A239</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_1"><i><font color="Green" title="E82">A244</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_1_1_1_4"><i><font color="Green" title="E83">A251</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1_1_1_1_1_1_4"><i><font color="Green" title="E84">A252</font></i></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:33_1_1_7_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_2"/><i><font color="Green" title="E79">A253</font></i>: 
(  <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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/><div class="add"><b>then consider </b><font color="Maroon" title="c16">i</font>, <font color="Maroon" title="c17">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> <b> such that </b><br/><a NAME="E3:33_1_1_7_1_3_2_1_1_1_1_2"/><i><font color="Green" title="E80">A254</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="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="c16">i</font>,<font color="Maroon" title="c17">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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>and </b><br/><a NAME="E4:33_1_1_7_1_3_2_1_1_1_1_2"/><i><font color="Green" title="E81">A255</font></i>: 
<font 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#K1" title="NAT_1:func.1">+</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="c16">i</font>,<font color="Maroon" title="c17">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1"><i><font color="Green" title="E35">A80</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3"><i><font color="Green" title="E52">A170</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_7_1_3"><i><font color="Green" title="E61">A179</font></i></a></i>;<br/><a NAME="E5:33_1_1_7_1_3_2_1_1_1_1_2"/><i><font color="Green" title="E82">A256</font></i>: 
( <span class="p1">(<span class="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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"><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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_7_1_3"><i><font color="Green" title="E59">A177</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3_2_1_1_1_1_2"><i><font color="Green" title="E81">A255</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_7_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_7_1_3"><i><font color="Green" title="E60">A178</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3_2_1_1_1_1_2"><i><font color="Green" title="E80">A254</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3_2_1_1_1_1_2"><i><font color="Green" title="E81">A255</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_2"><i><font color="Green" title="E82">A256</font></i></a>, <a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_2_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E83">A257</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_2_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E84">A258</font></i>: 
<span class="p1">(<span class="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E3:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E83">A257</font></i></a>, <a class="ref" href="gobrd13.html#T35" title="GOBRD13:th.35">GOBRD13:35</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_7_1_3"><i><font color="Green" title="E66">A184</font></i></a>, <a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E76">A238</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_2"><i><font color="Green" title="E79">A253</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_2"><i><font color="Green" title="E82">A256</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E83">A257</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E84">A258</font></i></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></i>;<br/><a NAME="E5:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E83">A257</font></i></a>, <a class="ref" href="gobrd13.html#T36" title="GOBRD13:th.36">GOBRD13:36</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_7_1_3"><i><font color="Green" title="E66">A184</font></i></a>, <a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E76">A238</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_2"><i><font color="Green" title="E79">A253</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_2"><i><font color="Green" title="E82">A256</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E83">A257</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E84">A258</font></i></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_2_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E83">A259</font></i>: 
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_2_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E84">A260</font></i>: 
<span class="p1">(<span class="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 ;<br/><div class="add"><a NAME="E3:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_2_1_1_2"><i><font color="Green" title="E83">A259</font></i></a>, <a class="ref" href="gobrd13.html#T37" title="GOBRD13:th.37">GOBRD13:37</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E19:33_1_1_7_1_3"><i><font color="Green" title="E67">A185</font></i></a>, <a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E76">A238</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_2"><i><font color="Green" title="E79">A253</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_2"><i><font color="Green" title="E82">A256</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_2_1_1_2"><i><font color="Green" title="E83">A259</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1_1_1_2_1_1_2"><i><font color="Green" title="E84">A260</font></i></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></i>;<br/><a NAME="E5:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_2_1_1_2"><i><font color="Green" title="E83">A259</font></i></a>, <a class="ref" href="gobrd13.html#T38" title="GOBRD13:th.38">GOBRD13:38</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E19:33_1_1_7_1_3"><i><font color="Green" title="E67">A185</font></i></a>, <a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E76">A238</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_2"><i><font color="Green" title="E79">A253</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_2"><i><font color="Green" title="E82">A256</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_2_1_1_2"><i><font color="Green" title="E83">A259</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1_1_1_2_1_1_2"><i><font color="Green" title="E84">A260</font></i></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_2_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_2_1_1_3"/><i><font color="Green" title="E83">A261</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_2_1_1_1_1_2_1_1_3"/><i><font color="Green" title="E84">A262</font></i>: 
<span class="p1">(<span class="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 ;<br/><div class="add"><a NAME="E3:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_2_1_1_3"><i><font color="Green" title="E83">A261</font></i></a>, <a class="ref" href="gobrd13.html#T39" title="GOBRD13:th.39">GOBRD13:39</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_7_1_3"><i><font color="Green" title="E68">A186</font></i></a>, <a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E76">A238</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E77">A239</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_2"><i><font color="Green" title="E79">A253</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_2"><i><font color="Green" title="E82">A256</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_2_1_1_3"><i><font color="Green" title="E83">A261</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1_1_1_2_1_1_3"><i><font color="Green" title="E84">A262</font></i></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></i>;<br/><a NAME="E5:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_2_1_1_3"><i><font color="Green" title="E83">A261</font></i></a>, <a class="ref" href="gobrd13.html#T40" title="GOBRD13:th.40">GOBRD13:40</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_7_1_3"><i><font color="Green" title="E68">A186</font></i></a>, <a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E76">A238</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E77">A239</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_2"><i><font color="Green" title="E79">A253</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_2"><i><font color="Green" title="E82">A256</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_2_1_1_3"><i><font color="Green" title="E83">A261</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1_1_1_2_1_1_3"><i><font color="Green" title="E84">A262</font></i></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_2_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_2_1_1_4"/><i><font color="Green" title="E83">A263</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_2_1_1_1_1_2_1_1_4"/><i><font color="Green" title="E84">A264</font></i>: 
<span class="p1">(<span class="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E3:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_2_1_1_4"><i><font color="Green" title="E83">A263</font></i></a>, <a class="ref" href="gobrd13.html#T41" title="GOBRD13:th.41">GOBRD13:41</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E21:33_1_1_7_1_3"><i><font color="Green" title="E69">A187</font></i></a>, <a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E76">A238</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E78">A240</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_2"><i><font color="Green" title="E79">A253</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_2"><i><font color="Green" title="E82">A256</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_2_1_1_4"><i><font color="Green" title="E83">A263</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1_1_1_2_1_1_4"><i><font color="Green" title="E84">A264</font></i></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></i>;<br/><a NAME="E5:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_2_1_1_4"><i><font color="Green" title="E83">A263</font></i></a>, <a class="ref" href="gobrd13.html#T42" title="GOBRD13:th.42">GOBRD13:42</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E21:33_1_1_7_1_3"><i><font color="Green" title="E69">A187</font></i></a>, <a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E76">A238</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E78">A240</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_2"><i><font color="Green" title="E79">A253</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_2"><i><font color="Green" title="E82">A256</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_2_1_1_4"><i><font color="Green" title="E83">A263</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1_1_1_2_1_1_4"><i><font color="Green" title="E84">A264</font></i></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:33_1_1_7_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_3"/><i><font color="Green" title="E79">A265</font></i>: 
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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/><div class="add"><b>then consider </b><font color="Maroon" title="c16">i</font>, <font color="Maroon" title="c17">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> <b> such that </b><br/><a NAME="E3:33_1_1_7_1_3_2_1_1_1_1_3"/><i><font color="Green" title="E80">A266</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="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="c16">i</font>,<font color="Maroon" title="c17">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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>and </b><br/><a NAME="E4:33_1_1_7_1_3_2_1_1_1_1_3"/><i><font color="Green" title="E81">A267</font></i>: 
<font 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#K1" title="NAT_1:func.1">+</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="c16">i</font>,<font color="Maroon" title="c17">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1"><i><font color="Green" title="E35">A80</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3"><i><font color="Green" title="E52">A170</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_7_1_3"><i><font color="Green" title="E61">A179</font></i></a></i>;<br/><a NAME="E5:33_1_1_7_1_3_2_1_1_1_1_3"/><i><font color="Green" title="E82">A268</font></i>: 
( <span class="p1">(<span class="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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"><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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_7_1_3"><i><font color="Green" title="E59">A177</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3_2_1_1_1_1_3"><i><font color="Green" title="E81">A267</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_7_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> ) or ( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_7_1_3"><i><font color="Green" title="E60">A178</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3_2_1_1_1_1_3"><i><font color="Green" title="E80">A266</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3_2_1_1_1_1_3"><i><font color="Green" title="E81">A267</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_3"><i><font color="Green" title="E82">A268</font></i></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_3_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_3_1_1_1"/><i><font color="Green" title="E83">A269</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_2_1_1_1_1_3_1_1_1"/><i><font color="Green" title="E84">A270</font></i>: 
<span class="p1">(<span class="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 ;<br/><div class="add"><a NAME="E3:33_1_1_7_1_3_2_1_1_1_1_3_1_1_1"/>
( <font color="Maroon" title="c14">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_7_1_3"><i><font color="Green" title="E61">A179</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_3_1_1_1"><i><font color="Green" title="E83">A269</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_7_1_3"><i><font color="Green" title="E70">A188</font></i></a>, <a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E76">A238</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E77">A239</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_3"><i><font color="Green" title="E79">A265</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_3"><i><font color="Green" title="E82">A268</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_3_1_1_1"><i><font color="Green" title="E83">A269</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1_1_1_3_1_1_1"><i><font color="Green" title="E84">A270</font></i></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></i>;<br/><a NAME="E5:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_3_1_1_1"><i><font color="Green" title="E83">A269</font></i></a>, <a class="ref" href="gobrd13.html#T35" title="GOBRD13:th.35">GOBRD13:35</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_7_1_3"><i><font color="Green" title="E70">A188</font></i></a>, <a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E76">A238</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E77">A239</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_3"><i><font color="Green" title="E79">A265</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_3"><i><font color="Green" title="E82">A268</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_3_1_1_1"><i><font color="Green" title="E83">A269</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1_1_1_3_1_1_1"><i><font color="Green" title="E84">A270</font></i></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_3_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_3_1_1_2"/><i><font color="Green" title="E83">A271</font></i>: 
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_2_1_1_1_1_3_1_1_2"/><i><font color="Green" title="E84">A272</font></i>: 
<span class="p1">(<span class="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E3:33_1_1_7_1_3_2_1_1_1_1_3_1_1_2"/>
( <font color="Maroon" title="c13">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_7_1_3"><i><font color="Green" title="E61">A179</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_3_1_1_2"><i><font color="Green" title="E83">A271</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_7_1_3"><i><font color="Green" title="E71">A189</font></i></a>, <a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E76">A238</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_3"><i><font color="Green" title="E79">A265</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_3"><i><font color="Green" title="E82">A268</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_3_1_1_2"><i><font color="Green" title="E83">A271</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1_1_1_3_1_1_2"><i><font color="Green" title="E84">A272</font></i></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></i>;<br/><a NAME="E5:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_3_1_1_2"><i><font color="Green" title="E83">A271</font></i></a>, <a class="ref" href="gobrd13.html#T37" title="GOBRD13:th.37">GOBRD13:37</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_7_1_3"><i><font color="Green" title="E71">A189</font></i></a>, <a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E76">A238</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_3"><i><font color="Green" title="E79">A265</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_3"><i><font color="Green" title="E82">A268</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_3_1_1_2"><i><font color="Green" title="E83">A271</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1_1_1_3_1_1_2"><i><font color="Green" title="E84">A272</font></i></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_3_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_3_1_1_3"/><i><font color="Green" title="E83">A273</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_2_1_1_1_1_3_1_1_3"/><i><font color="Green" title="E84">A274</font></i>: 
<span class="p1">(<span class="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E3:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_7_1_3"><i><font color="Green" title="E61">A179</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_3_1_1_3"><i><font color="Green" title="E83">A273</font></i></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_7_1_3"><i><font color="Green" title="E72">A190</font></i></a>, <a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E76">A238</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E78">A240</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_3"><i><font color="Green" title="E79">A265</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_3"><i><font color="Green" title="E82">A268</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_3_1_1_3"><i><font color="Green" title="E83">A273</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1_1_1_3_1_1_3"><i><font color="Green" title="E84">A274</font></i></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></i>;<br/><a NAME="E5:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_3_1_1_3"><i><font color="Green" title="E83">A273</font></i></a>, <a class="ref" href="gobrd13.html#T39" title="GOBRD13:th.39">GOBRD13:39</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_7_1_3"><i><font color="Green" title="E72">A190</font></i></a>, <a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E76">A238</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E78">A240</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_3"><i><font color="Green" title="E79">A265</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_3"><i><font color="Green" title="E82">A268</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_3_1_1_3"><i><font color="Green" title="E83">A273</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1_1_1_3_1_1_3"><i><font color="Green" title="E84">A274</font></i></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_3_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_3_1_1_4"/><i><font color="Green" title="E83">A275</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_2_1_1_1_1_3_1_1_4"/><i><font color="Green" title="E84">A276</font></i>: 
<span class="p1">(<span class="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j2</font>
 ;<br/><div class="add"><a NAME="E3:33_1_1_7_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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_7_1_3"><i><font color="Green" title="E61">A179</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_3_1_1_4"><i><font color="Green" title="E83">A275</font></i></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E25:33_1_1_7_1_3"><i><font color="Green" title="E73">A191</font></i></a>, <a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E76">A238</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_3"><i><font color="Green" title="E79">A265</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_3"><i><font color="Green" title="E82">A268</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_3_1_1_4"><i><font color="Green" title="E83">A275</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1_1_1_3_1_1_4"><i><font color="Green" title="E84">A276</font></i></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></i>;<br/><a NAME="E5:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3"><i><font color="Green" title="E54">A172</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7_1_3"><i><font color="Green" title="E55">A173</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_3_1_1_4"><i><font color="Green" title="E83">A275</font></i></a>, <a class="ref" href="gobrd13.html#T41" title="GOBRD13:th.41">GOBRD13:41</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_7_1_3"><i><font color="Green" title="E57">A175</font></i></a>, <a class="txt" href="jordan9.html#E25:33_1_1_7_1_3"><i><font color="Green" title="E73">A191</font></i></a>, <a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1"><i><font color="Green" title="E76">A238</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_3"><i><font color="Green" title="E79">A265</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_1_1_1_3"><i><font color="Green" title="E82">A268</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_1_1_1_3_1_1_4"><i><font color="Green" title="E83">A275</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_7_1_3_2_1_1_1_1_3_1_1_4"><i><font color="Green" title="E84">A276</font></i></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:33_1_1_7_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3_2_1_2"/>
<font color="Maroon" title="c15">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:33_1_1_7_1_3_2_1_2"/><b>then </b>
<font color="Maroon" title="c15">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_2_1_2"/><b>then </b><i><font color="Green" title="E76">A277</font></i>: 
<font color="Maroon" title="c15">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_7"><i><font color="Green" title="E49">A136</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then consider </b><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> <b> such that </b><br/><a NAME="E5:33_1_1_7_1_3_2_1_2"/><i><font color="Green" title="E77">A278</font></i>: 
( <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> &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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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> )
 <b>and </b><br/><a NAME="E6:33_1_1_7_1_3_2_1_2"/><i><font color="Green" title="E78">A279</font></i>: 
( <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> &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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 <b>and </b><br/><a NAME="E7:33_1_1_7_1_3_2_1_2"/><i><font color="Green" title="E79">A280</font></i>: 
( ( <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E8:33_1_1_7_1_3_2_1_2"/><i><font color="Green" title="E80">A281</font></i>: 
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#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E9:33_1_1_7_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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E76">A277</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E10:33_1_1_7_1_3_2_1_2"/><b>then </b><i><font color="Green" title="E81">A282</font></i>: 
( <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> &amp; <font color="Maroon" title="c15">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E76">A277</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E80">A281</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:33_1_1_7_1_3_2_1_2"/><i><font color="Green" title="E82">A283</font></i>: 
(  <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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_7"><i><font color="Green" title="E47">A134</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E76">A277</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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="33_1_1_7_1_3_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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> )
 ;<br/><div class="add"><b>then consider </b><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> <b> such that </b><br/><a NAME="E3:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>and </b><br/><a NAME="E4:33_1_1_7_1_3_2_1_2_1_1_1"/><i><font color="Green" title="E83">A284</font></i>: 
<font 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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1"><i><font color="Green" title="E35">A80</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3"><i><font color="Green" title="E52">A170</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_7_1_3"><i><font color="Green" title="E61">A179</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c22">i</font> = <font color="Maroon" title="c20">i</font>;<br/><b>take </b><font color="Maroon" title="c23">j</font> = <font color="Maroon" title="c21">j</font>;<br/><b>thus </b><a NAME="E5:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_7_1_3_2_1_2_1_1_1"><i><font color="Green" title="E83">A284</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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/><div class="add"><b>then consider </b><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> <b> such that </b><br/><a NAME="E3:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>and </b><br/><a NAME="E4:33_1_1_7_1_3_2_1_2_1_1_2"/><i><font color="Green" title="E83">A285</font></i>: 
<font 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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1"><i><font color="Green" title="E35">A80</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3"><i><font color="Green" title="E52">A170</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_7_1_3"><i><font color="Green" title="E61">A179</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c22">i</font> = <font color="Maroon" title="c20">i</font>;<br/><b>take </b><font color="Maroon" title="c23">j</font> = <font color="Maroon" title="c21">j</font>;<br/><b>thus </b><a NAME="E5:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_7_1_3_2_1_2_1_1_2"><i><font color="Green" title="E83">A285</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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/><div class="add"><b>then consider </b><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> <b> such that </b><br/><a NAME="E3:33_1_1_7_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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>and </b><br/><a NAME="E4:33_1_1_7_1_3_2_1_2_1_1_3"/><i><font color="Green" title="E83">A286</font></i>: 
<font 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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1"><i><font color="Green" title="E35">A80</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7"><i><font color="Green" title="E48">A135</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3"><i><font color="Green" title="E52">A170</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_7_1_3"><i><font color="Green" title="E61">A179</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c22">i</font> = <font color="Maroon" title="c20">i</font>;<br/><b>take </b><font color="Maroon" title="c23">j</font> = <font color="Maroon" title="c21">j</font>;<br/><b>thus </b><a NAME="E5:33_1_1_7_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_7_1_3_2_1_2_1_1_3"><i><font color="Green" title="E83">A286</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><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> <b> such that </b><br/><a NAME="E14:33_1_1_7_1_3_2_1_2"/><i><font color="Green" title="E83">A287</font></i>: 
<font 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#K1" title="NAT_1:func.1">+</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>
 ;<br/><a NAME="E15:33_1_1_7_1_3_2_1_2"/><i><font color="Green" title="E84">A288</font></i>: 
( <span class="p1">(<span class="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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> &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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E77">A278</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E78">A279</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E81">A282</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E83">A287</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_7_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E79">A280</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3_2_1_2_2_1_1"/><i><font color="Green" title="E85">A289</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_7_1_3_2_1_2_2_1_1"/><b>then </b>
(  <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="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E76">A277</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E77">A278</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E78">A279</font></i></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></i>;<br/><b>hence </b><a NAME="E3:33_1_1_7_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E77">A278</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E78">A279</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E82">A283</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E84">A288</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_2_2_1_1"><i><font color="Green" title="E85">A289</font></i></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></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3_2_1_2_2_1_2"/><i><font color="Green" title="E85">A290</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_7_1_3_2_1_2_2_1_2"/><b>then </b>
(  <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E76">A277</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E77">A278</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E78">A279</font></i></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></i>;<br/><b>hence </b><a NAME="E3:33_1_1_7_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E77">A278</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E78">A279</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E82">A283</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E84">A288</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_2_2_1_2"><i><font color="Green" title="E85">A290</font></i></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></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_2_2_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3_2_1_2_2_1_3"/><i><font color="Green" title="E85">A291</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_7_1_3_2_1_2_2_1_3"/><b>then </b>
(  <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="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E76">A277</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E77">A278</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E78">A279</font></i></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></i>;<br/><b>hence </b><a NAME="E3:33_1_1_7_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E77">A278</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E78">A279</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E82">A283</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E84">A288</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_2_2_1_3"><i><font color="Green" title="E85">A291</font></i></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></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_2_2_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3_2_1_2_2_1_4"/><i><font color="Green" title="E85">A292</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_7_1_3_2_1_2_2_1_4"/><b>then </b>
(  <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j2</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_7"><i><font color="Green" title="E46">A133</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E76">A277</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E77">A278</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E78">A279</font></i></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></i>;<br/><b>hence </b><a NAME="E3:33_1_1_7_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E26:33_1_1_7_1_3"><i><font color="Green" title="E74">A192</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_7_1_3"><i><font color="Green" title="E75">A237</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E77">A278</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E78">A279</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E82">A283</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_7_1_3_2_1_2"><i><font color="Green" title="E84">A288</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_7_1_3_2_1_2_2_1_4"><i><font color="Green" title="E85">A292</font></i></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></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E17:33_1_1_7_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E29:33_1_1_7_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E19:33_1_1"/><i><font color="Green" title="E46">A293</font></i>: 
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>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="jordan9.html#E16:33_1_1"><i><font color="Green" title="E43">A96</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1"><i><font color="Green" title="E45">A132</font></i></a>);<br/></i>
<a NAME="E20:33_1_1"/><i><font color="Green" title="E47">A294</font></i>: 
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="finseq_4.html#K4" title="FINSEQ_4: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"><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="binarith.html#K3" title="BINARITH:func.3">-'</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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="33_1_1_8"><b>proof </b></a><div class="add">

<b>let </b><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> ;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_8"/><i><font color="Green" title="E48">A295</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>and </b><br/><a NAME="E2:33_1_1_8"/><i><font color="Green" title="E49">A296</font></i>: 
( <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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>and </b><br/><a NAME="E3:33_1_1_8"/><i><font color="Green" title="E50">A297</font></i>: 
( <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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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> )
 ;<br/>

<a NAME="E4:33_1_1_8"/><i><font color="Green" title="E51">A298</font></i>: 
<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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 
<b>by </b><i><a class="ref" href="binarith.html#T52" title="BINARITH:th.52">BINARITH:52</a></i>;<br/>
<a NAME="E5:33_1_1_8"/><i><font color="Green" title="E52">A299</font></i>: 
<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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a></i>;<br/>
<a NAME="E6:33_1_1_8"/><i><font color="Green" title="E53">A300</font></i>: 
 <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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a></i>;<br/>
<a NAME="E7:33_1_1_8"/><b>then </b><i><font color="Green" title="E54">A301</font></i>: 
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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_8"><i><font color="Green" title="E48">A295</font></i></a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a></i>;<br/>
<a NAME="E8:33_1_1_8"/><i><font color="Green" title="E55">A302</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_8"><i><font color="Green" title="E48">A295</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_8"><i><font color="Green" title="E53">A300</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/>
<a NAME="E9:33_1_1_8"/><b>then </b><i><font color="Green" title="E56">A303</font></i>: 
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_8"><i><font color="Green" title="E54">A301</font></i></a></i>;<br/>
<a NAME="E10:33_1_1_8"/><i><font color="Green" title="E57">A304</font></i>: 
( <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="binarith.html#K3" title="BINARITH:func.3">-'</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> &amp;  <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> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_8"><i><font color="Green" title="E48">A295</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_8"><i><font color="Green" title="E51">A298</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_8"><i><font color="Green" title="E53">A300</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_8"><i><font color="Green" title="E54">A301</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:33_1_1_8"/><i><font color="Green" title="E58">A305</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan9.html#E8:33_1_1_8"><i><font color="Green" title="E55">A302</font></i></a></i>;<br/>
<a NAME="E12:33_1_1_8"/><i><font color="Green" title="E59">A306</font></i>: 
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">j2</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:33_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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> )
 ;<br/><b>then consider </b><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> <b> such that </b><br/><a NAME="E3:33_1_1_8_1"/><i><font color="Green" title="E60">A307</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>and </b><br/><a NAME="E4:33_1_1_8_1"/><i><font color="Green" title="E61">A308</font></i>: 
<font 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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1"><i><font color="Green" title="E35">A80</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_8"><i><font color="Green" title="E48">A295</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_8"><i><font color="Green" title="E52">A299</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_8"><i><font color="Green" title="E53">A300</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_8"><i><font color="Green" title="E56">A303</font></i></a></i>;<br/><a NAME="E5:33_1_1_8_1"/><i><font color="Green" title="E62">A309</font></i>: 
( <span class="p1">(<span class="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_8"><i><font color="Green" title="E49">A296</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_8"><i><font color="Green" title="E50">A297</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_8"><i><font color="Green" title="E57">A304</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_8_1"><i><font color="Green" title="E61">A308</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><a NAME="E6:33_1_1_8_1"/><i><font color="Green" title="E63">A310</font></i>: 
<span class="p1">(<span class="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_8_1"><i><font color="Green" title="E61">A308</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></i>;<br/><b>thus </b><a NAME="E7:33_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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_8_1"><i><font color="Green" title="E60">A307</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_8_1"><i><font color="Green" title="E61">A308</font></i></a></i>;<br/><b>thus </b><a NAME="E8:33_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_8"><i><font color="Green" title="E49">A296</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_8"><i><font color="Green" title="E50">A297</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_8"><i><font color="Green" title="E55">A302</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_8"><i><font color="Green" title="E58">A305</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_8"><i><font color="Green" title="E59">A306</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_8_1"><i><font color="Green" title="E60">A307</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_8_1"><i><font color="Green" title="E61">A308</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_8_1"><i><font color="Green" title="E62">A309</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_8_1"><i><font color="Green" title="E63">A310</font></i></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></i>;<br/><b>thus </b><a NAME="E9:33_1_1_8_1"/>
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_8"><i><font color="Green" title="E49">A296</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_8"><i><font color="Green" title="E50">A297</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_8"><i><font color="Green" title="E55">A302</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_8"><i><font color="Green" title="E58">A305</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_8"><i><font color="Green" title="E59">A306</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_8_1"><i><font color="Green" title="E60">A307</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_8_1"><i><font color="Green" title="E61">A308</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_8_1"><i><font color="Green" title="E62">A309</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_8_1"><i><font color="Green" title="E63">A310</font></i></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></i>;<br/><b>thus </b><a NAME="E10:33_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_8"><i><font color="Green" title="E49">A296</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_8"><i><font color="Green" title="E50">A297</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_8"><i><font color="Green" title="E55">A302</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_8"><i><font color="Green" title="E58">A305</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_8"><i><font color="Green" title="E59">A306</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_8_1"><i><font color="Green" title="E60">A307</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_8_1"><i><font color="Green" title="E61">A308</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_8_1"><i><font color="Green" title="E62">A309</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_8_1"><i><font color="Green" title="E63">A310</font></i></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></i>;<br/><b>thus </b><a NAME="E11:33_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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_8"><i><font color="Green" title="E49">A296</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_8"><i><font color="Green" title="E50">A297</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_8"><i><font color="Green" title="E55">A302</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_8"><i><font color="Green" title="E58">A305</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_8"><i><font color="Green" title="E59">A306</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_8_1"><i><font color="Green" title="E60">A307</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_8_1"><i><font color="Green" title="E61">A308</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_8_1"><i><font color="Green" title="E62">A309</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_8_1"><i><font color="Green" title="E63">A310</font></i></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:33_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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/><b>then consider </b><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> <b> such that </b><br/><a NAME="E3:33_1_1_8_2"/><i><font color="Green" title="E60">A311</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>and </b><br/><a NAME="E4:33_1_1_8_2"/><i><font color="Green" title="E61">A312</font></i>: 
<font 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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1"><i><font color="Green" title="E35">A80</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_8"><i><font color="Green" title="E48">A295</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_8"><i><font color="Green" title="E52">A299</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_8"><i><font color="Green" title="E53">A300</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_8"><i><font color="Green" title="E56">A303</font></i></a></i>;<br/><a NAME="E5:33_1_1_8_2"/><i><font color="Green" title="E62">A313</font></i>: 
( <span class="p1">(<span class="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_8"><i><font color="Green" title="E49">A296</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_8"><i><font color="Green" title="E50">A297</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_8"><i><font color="Green" title="E57">A304</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_8_2"><i><font color="Green" title="E61">A312</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><a NAME="E6:33_1_1_8_2"/><i><font color="Green" title="E63">A314</font></i>: 
<span class="p1">(<span class="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_8_2"><i><font color="Green" title="E61">A312</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></i>;<br/><b>thus </b><a NAME="E7:33_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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_8_2"><i><font color="Green" title="E60">A311</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_8_2"><i><font color="Green" title="E61">A312</font></i></a></i>;<br/><b>thus </b><a NAME="E8:33_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_8"><i><font color="Green" title="E49">A296</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_8"><i><font color="Green" title="E50">A297</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_8"><i><font color="Green" title="E55">A302</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_8"><i><font color="Green" title="E58">A305</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_8"><i><font color="Green" title="E59">A306</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_8_2"><i><font color="Green" title="E60">A311</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_8_2"><i><font color="Green" title="E61">A312</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_8_2"><i><font color="Green" title="E62">A313</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_8_2"><i><font color="Green" title="E63">A314</font></i></a>, <a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></i>;<br/><b>thus </b><a NAME="E9:33_1_1_8_2"/>
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_8"><i><font color="Green" title="E49">A296</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_8"><i><font color="Green" title="E50">A297</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_8"><i><font color="Green" title="E55">A302</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_8"><i><font color="Green" title="E58">A305</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_8"><i><font color="Green" title="E59">A306</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_8_2"><i><font color="Green" title="E60">A311</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_8_2"><i><font color="Green" title="E61">A312</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_8_2"><i><font color="Green" title="E62">A313</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_8_2"><i><font color="Green" title="E63">A314</font></i></a>, <a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></i>;<br/><b>thus </b><a NAME="E10:33_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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_8"><i><font color="Green" title="E49">A296</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_8"><i><font color="Green" title="E50">A297</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_8"><i><font color="Green" title="E55">A302</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_8"><i><font color="Green" title="E58">A305</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_8"><i><font color="Green" title="E59">A306</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_8_2"><i><font color="Green" title="E60">A311</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_8_2"><i><font color="Green" title="E61">A312</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_8_2"><i><font color="Green" title="E62">A313</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_8_2"><i><font color="Green" title="E63">A314</font></i></a>, <a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></i>;<br/><b>thus </b><a NAME="E11:33_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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_8"><i><font color="Green" title="E49">A296</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_8"><i><font color="Green" title="E50">A297</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_8"><i><font color="Green" title="E55">A302</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_8"><i><font color="Green" title="E58">A305</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_8"><i><font color="Green" title="E59">A306</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_8_2"><i><font color="Green" title="E60">A311</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_8_2"><i><font color="Green" title="E61">A312</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_8_2"><i><font color="Green" title="E62">A313</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_8_2"><i><font color="Green" title="E63">A314</font></i></a>, <a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E15:33_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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/>

<b>then consider </b><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> <b> such that </b><br/><a NAME="E17:33_1_1_8"/><i><font color="Green" title="E60">A315</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>and </b><br/><a NAME="E18:33_1_1_8"/><i><font color="Green" title="E61">A316</font></i>: 
<font 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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1"><i><font color="Green" title="E35">A80</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_8"><i><font color="Green" title="E48">A295</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_8"><i><font color="Green" title="E52">A299</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_8"><i><font color="Green" title="E53">A300</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_8"><i><font color="Green" title="E56">A303</font></i></a></i>;<br/>
<a NAME="E19:33_1_1_8"/><i><font color="Green" title="E62">A317</font></i>: 
( <span class="p1">(<span class="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_8"><i><font color="Green" title="E49">A296</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_8"><i><font color="Green" title="E50">A297</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_8"><i><font color="Green" title="E57">A304</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_8"><i><font color="Green" title="E61">A316</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E20:33_1_1_8"/><i><font color="Green" title="E63">A318</font></i>: 
<span class="p1">(<span class="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <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#K1" title="NAT_1:func.1">+</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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E18:33_1_1_8"><i><font color="Green" title="E61">A316</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></i>;<br/>
<b>thus </b><a NAME="E21:33_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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E17:33_1_1_8"><i><font color="Green" title="E60">A315</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_8"><i><font color="Green" title="E61">A316</font></i></a></i>;<br/>

<b>thus </b><a NAME="E22:33_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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_8"><i><font color="Green" title="E49">A296</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_8"><i><font color="Green" title="E50">A297</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_8"><i><font color="Green" title="E55">A302</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_8"><i><font color="Green" title="E58">A305</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_8"><i><font color="Green" title="E59">A306</font></i></a>, <a class="txt" href="jordan9.html#E17:33_1_1_8"><i><font color="Green" title="E60">A315</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_8"><i><font color="Green" title="E61">A316</font></i></a>, <a class="txt" href="jordan9.html#E19:33_1_1_8"><i><font color="Green" title="E62">A317</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_8"><i><font color="Green" title="E63">A318</font></i></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></i>;<br/>

<b>thus </b><a NAME="E23:33_1_1_8"/>
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_8"><i><font color="Green" title="E49">A296</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_8"><i><font color="Green" title="E50">A297</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_8"><i><font color="Green" title="E55">A302</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_8"><i><font color="Green" title="E58">A305</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_8"><i><font color="Green" title="E59">A306</font></i></a>, <a class="txt" href="jordan9.html#E17:33_1_1_8"><i><font color="Green" title="E60">A315</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_8"><i><font color="Green" title="E61">A316</font></i></a>, <a class="txt" href="jordan9.html#E19:33_1_1_8"><i><font color="Green" title="E62">A317</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_8"><i><font color="Green" title="E63">A318</font></i></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></i>;<br/>

<b>thus </b><a NAME="E24:33_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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_8"><i><font color="Green" title="E49">A296</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_8"><i><font color="Green" title="E50">A297</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_8"><i><font color="Green" title="E55">A302</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_8"><i><font color="Green" title="E58">A305</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_8"><i><font color="Green" title="E59">A306</font></i></a>, <a class="txt" href="jordan9.html#E17:33_1_1_8"><i><font color="Green" title="E60">A315</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_8"><i><font color="Green" title="E61">A316</font></i></a>, <a class="txt" href="jordan9.html#E19:33_1_1_8"><i><font color="Green" title="E62">A317</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_8"><i><font color="Green" title="E63">A318</font></i></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></i>;<br/>

<b>thus </b><a NAME="E25:33_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_8"><i><font color="Green" title="E49">A296</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_8"><i><font color="Green" title="E50">A297</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_8"><i><font color="Green" title="E55">A302</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_8"><i><font color="Green" title="E58">A305</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_8"><i><font color="Green" title="E59">A306</font></i></a>, <a class="txt" href="jordan9.html#E17:33_1_1_8"><i><font color="Green" title="E60">A315</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_8"><i><font color="Green" title="E61">A316</font></i></a>, <a class="txt" href="jordan9.html#E19:33_1_1_8"><i><font color="Green" title="E62">A317</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_8"><i><font color="Green" title="E63">A318</font></i></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:33_1_1"/><i><font color="Green" title="E48">A319</font></i>: 
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#K1" title="NAT_1:func.1">+</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="33_1_1_9"><b>proof </b></a><div class="add">

<b>let </b><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> ;<br/>

<a NAME="E1:33_1_1_9"/><i><font color="Green" title="E49">A320</font></i>: 
<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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a></i>;<br/>
<a NAME="E2:33_1_1_9"/><i><font color="Green" title="E50">A321</font></i>: 
 <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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a></i>;<br/>
<a NAME="E3:33_1_1_9"/><i><font color="Green" title="E51">A322</font></i>: 
(  <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#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3 &amp;  <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> )
 
<b>by </b><i><a class="ref" href="jordan8.html#D1" title="JORDAN8:def.1">JORDAN8:def 1</a></i>;<br/>
<a NAME="E4:33_1_1_9"/>
 <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
 
<b>by </b><i><a class="ref" href="jordan8.html#T13" title="JORDAN8:th.13">JORDAN8:13</a></i>;<br/>
<a NAME="E5:33_1_1_9"/><b>then </b><i><font color="Green" title="E52">A323</font></i>: 
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>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_9_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><b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_1"/>
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_1"/><b>then </b><i><font color="Green" title="E53">A324</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><b>consider </b><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> <b> such that </b><br/><a NAME="E4:33_1_1_9_1_1"/><i><font color="Green" title="E54">A325</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 <b>and </b><br/><a NAME="E5:33_1_1_9_1_1"/>
(  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 <b>and </b><br/><a NAME="E6:33_1_1_9_1_1"/><i><font color="Green" title="E55">A326</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1"><i><font color="Green" title="E35">A80</font></i></a></i>;<br/><b>take </b>
<font color="Maroon" title="c11">i</font>
;<br/><b>take </b><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>;<br/><a NAME="E7:33_1_1_9_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>
 <b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_9_1_1"><i><font color="Green" title="E54">A325</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E8:33_1_1_9_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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_9"><i><font color="Green" title="E51">A322</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_9"><i><font color="Green" title="E52">A323</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_9_1_1"><i><font color="Green" title="E54">A325</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E9:33_1_1_9_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1"><i><font color="Green" title="E34">A79</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_9_1_1"><i><font color="Green" title="E53">A324</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_9_1_1"><i><font color="Green" title="E55">A326</font></i></a>, <a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_2"/><i><font color="Green" title="E53">A327</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>consider </b><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> <b> such that </b><br/><a NAME="E3:33_1_1_9_1_2"/><i><font color="Green" title="E54">A328</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 <b>and </b><br/><a NAME="E4:33_1_1_9_1_2"/><i><font color="Green" title="E55">A329</font></i>: 
(  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 <b>and </b><br/><a NAME="E5:33_1_1_9_1_2"/><i><font color="Green" title="E56">A330</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1"><i><font color="Green" title="E35">A80</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c12">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> <b> such that </b><br/><a NAME="E7:33_1_1_9_1_2"/><i><font color="Green" title="E57">A331</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i'</font> &amp; <font color="Maroon" title="c12">i'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K29" title="PSCOMP_1:func.29">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="c12">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">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="c12">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 <b>and </b><br/><a NAME="E8:33_1_1_9_1_2"/><i><font color="Green" title="E58">A332</font></i>: 
<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#K1" title="NAT_1:func.1">+</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="c12">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="c12">i'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1"><i><font color="Green" title="E35">A80</font></i></a></i>;<br/><a NAME="E9:33_1_1_9_1_2"/><i><font color="Green" title="E59">A333</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i'</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_9_1_2"><i><font color="Green" title="E54">A328</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_9_1_2"><i><font color="Green" title="E55">A329</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_9_1_2"><i><font color="Green" title="E57">A331</font></i></a>, <a class="ref" href="jordan9.html#T31" target="_self" title="JORDAN9:th.31">Th31</a></i>;<br/><b>take </b>
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
;<br/><b>take </b><font color="Maroon" title="c13">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>;<br/><a NAME="E10:33_1_1_9_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#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_9_1_2"><i><font color="Green" title="E54">A328</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E11:33_1_1_9_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c13">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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_9"><i><font color="Green" title="E51">A322</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_9"><i><font color="Green" title="E52">A323</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E12:33_1_1_9_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c13">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_9_1_2"><i><font color="Green" title="E53">A327</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_9_1_2"><i><font color="Green" title="E56">A330</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_9_1_2"><i><font color="Green" title="E58">A332</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_9_1_2"><i><font color="Green" title="E59">A333</font></i></a>, <a class="ref" href="finseq_1.html#D9" title="FINSEQ_1:def.9">FINSEQ_1:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_3"/><i><font color="Green" title="E53">A334</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_3"/><b>then </b><i><font color="Green" title="E54">A335</font></i>: 
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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_9"><i><font color="Green" title="E50">A321</font></i></a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a></i>;<br/><a NAME="E3:33_1_1_9_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"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_9"><i><font color="Green" title="E50">A321</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_9_1_3"><i><font color="Green" title="E53">A334</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><b>then consider </b><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> <b> such that </b><br/><a NAME="E5:33_1_1_9_1_3"/><i><font color="Green" title="E55">A336</font></i>: 
( <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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>and </b><br/><a NAME="E6:33_1_1_9_1_3"/><i><font color="Green" title="E56">A337</font></i>: 
( <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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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> )
 <b>and </b><br/><a NAME="E7:33_1_1_9_1_3"/><i><font color="Green" title="E57">A338</font></i>: 
( ( <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_9"><i><font color="Green" title="E49">A320</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_9_1_3"><i><font color="Green" title="E54">A335</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_9_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_9_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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="33_1_1_9_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_3_1_1_1"/><i><font color="Green" title="E58">A339</font></i>: 
(  <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_9_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_9_1_3"><i><font color="Green" title="E57">A338</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_9_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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_3_1_1_1_1_1_1"/><b>then </b>
( <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_9_1_3"><i><font color="Green" title="E53">A334</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_9_1_3"><i><font color="Green" title="E55">A336</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_9_1_3"><i><font color="Green" title="E56">A337</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_9_1_3_1_1_1"><i><font color="Green" title="E58">A339</font></i></a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_9_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#K1" title="NAT_1:func.1">+</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> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_3_1_1_1_1_1_2"/>
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_3_1_1_1_1_1_2"/><b>then </b>
( <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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_9_1_3"><i><font color="Green" title="E53">A334</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_9_1_3"><i><font color="Green" title="E55">A336</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_9_1_3"><i><font color="Green" title="E56">A337</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_9_1_3_1_1_1"><i><font color="Green" title="E58">A339</font></i></a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_9_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#K1" title="NAT_1:func.1">+</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> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_9_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#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_3_1_1_1_1_1_3"/><b>then </b>
( <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_9_1_3"><i><font color="Green" title="E53">A334</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_9_1_3"><i><font color="Green" title="E55">A336</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_9_1_3"><i><font color="Green" title="E56">A337</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_9_1_3_1_1_1"><i><font color="Green" title="E58">A339</font></i></a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_9_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#K1" title="NAT_1:func.1">+</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> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_9_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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_3_1_1_1_1_1_4"/><b>then </b>
( <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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_9_1_3"><i><font color="Green" title="E53">A334</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_9_1_3"><i><font color="Green" title="E55">A336</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_9_1_3"><i><font color="Green" title="E56">A337</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_9_1_3_1_1_1"><i><font color="Green" title="E58">A339</font></i></a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_9_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#K1" title="NAT_1:func.1">+</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> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_9_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#K1" title="NAT_1:func.1">+</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> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_3_1_1_2"/><i><font color="Green" title="E58">A340</font></i>: 
(  <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_9_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_9_1_3"><i><font color="Green" title="E57">A338</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_9_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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_3_1_1_2_1_1_1"/><b>then </b>
( <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_9_1_3"><i><font color="Green" title="E53">A334</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_9_1_3"><i><font color="Green" title="E55">A336</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_9_1_3"><i><font color="Green" title="E56">A337</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_9_1_3_1_1_2"><i><font color="Green" title="E58">A340</font></i></a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_9_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#K1" title="NAT_1:func.1">+</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> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_3_1_1_2_1_1_2"/>
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_3_1_1_2_1_1_2"/><b>then </b>
( <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_9_1_3"><i><font color="Green" title="E53">A334</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_9_1_3"><i><font color="Green" title="E55">A336</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_9_1_3"><i><font color="Green" title="E56">A337</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_9_1_3_1_1_2"><i><font color="Green" title="E58">A340</font></i></a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_9_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#K1" title="NAT_1:func.1">+</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> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_9_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#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_3_1_1_2_1_1_3"/><b>then </b>
( <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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_9_1_3"><i><font color="Green" title="E53">A334</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_9_1_3"><i><font color="Green" title="E55">A336</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_9_1_3"><i><font color="Green" title="E56">A337</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_9_1_3_1_1_2"><i><font color="Green" title="E58">A340</font></i></a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_9_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#K1" title="NAT_1:func.1">+</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> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_9_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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_3_1_1_2_1_1_4"/><b>then </b>
( <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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_9_1_3"><i><font color="Green" title="E53">A334</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_9_1_3"><i><font color="Green" title="E55">A336</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_9_1_3"><i><font color="Green" title="E56">A337</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_9_1_3_1_1_2"><i><font color="Green" title="E58">A340</font></i></a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_9_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#K1" title="NAT_1:func.1">+</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> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_9_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#K1" title="NAT_1:func.1">+</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> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_3_1_1_3"/><i><font color="Green" title="E58">A341</font></i>: 
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_9_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_9_1_3"><i><font color="Green" title="E57">A338</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_9_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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_3_1_1_3_1_1_1"/><b>then </b>
( <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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_9_1_3"><i><font color="Green" title="E53">A334</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_9_1_3"><i><font color="Green" title="E55">A336</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_9_1_3"><i><font color="Green" title="E56">A337</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_9_1_3_1_1_3"><i><font color="Green" title="E58">A341</font></i></a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_9_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#K1" title="NAT_1:func.1">+</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> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_3_1_1_3_1_1_2"/>
( <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_3_1_1_3_1_1_2"/><b>then </b>
( <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_9_1_3"><i><font color="Green" title="E53">A334</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_9_1_3"><i><font color="Green" title="E55">A336</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_9_1_3"><i><font color="Green" title="E56">A337</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_9_1_3_1_1_3"><i><font color="Green" title="E58">A341</font></i></a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_9_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#K1" title="NAT_1:func.1">+</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> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_9_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#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_3_1_1_3_1_1_3"/><b>then </b>
( <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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_9_1_3"><i><font color="Green" title="E53">A334</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_9_1_3"><i><font color="Green" title="E55">A336</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_9_1_3"><i><font color="Green" title="E56">A337</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_9_1_3_1_1_3"><i><font color="Green" title="E58">A341</font></i></a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_9_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#K1" title="NAT_1:func.1">+</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> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_3_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_9_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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_3_1_1_3_1_1_4"/><b>then </b>
( <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_9_1_3"><i><font color="Green" title="E53">A334</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_9_1_3"><i><font color="Green" title="E55">A336</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_9_1_3"><i><font color="Green" title="E56">A337</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_9_1_3_1_1_3"><i><font color="Green" title="E58">A341</font></i></a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_9_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#K1" title="NAT_1:func.1">+</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> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_9_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#K1" title="NAT_1:func.1">+</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> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:33_1_1_9_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#K1" title="NAT_1:func.1">+</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> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>5</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> ] <b>means </b>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="E22:33_1_1"/><i><font color="Green" title="E49">A342</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_10"><b>proof </b></a><div class="add">

<b>let </b><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> ;<br/>

<b>assume </b><a NAME="E1:33_1_1_10"/><i><font color="Green" title="E50">A343</font></i>: 
<font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 ;<br/>

<a NAME="E2:33_1_1_10"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> 0</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 0 <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> 0</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">m</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E3:33_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> 0</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1"><i><font color="Green" title="E34">A79</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_10"><i><font color="Green" title="E50">A343</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:33_1_1"/><i><font color="Green" title="E50">A344</font></i>: 
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>5</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_11"><b>proof </b></a><div class="add">

<b>let </b><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> ;<br/>

<b>assume </b><a NAME="E1:33_1_1_11"/><i><font color="Green" title="E51">A345</font></i>: 
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>
 ;<br/>

<b>let </b><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> ;<br/>

<b>assume </b><a NAME="E2:33_1_1_11"/><i><font color="Green" title="E52">A346</font></i>: 
<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#K1" title="NAT_1:func.1">+</a> 1
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_11_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_11_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_11"><i><font color="Green" title="E52">A346</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_11_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_11_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#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:33_1_1_11_1_1"/><b>then </b><i><font color="Green" title="E53">A347</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:33_1_1_11_1_1"/><i><font color="Green" title="E54">A348</font></i>: 
 <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>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c12">i</font>, <font color="Maroon" title="c13">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> <b> such that </b><br/><a NAME="E5:33_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="c12">i</font>,<font color="Maroon" title="c13">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>
 <b>and </b><br/><a NAME="E6:33_1_1_11_1_1"/><i><font color="Green" title="E55">A349</font></i>: 
<font 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#K1" title="NAT_1:func.1">+</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">i</font>,<font color="Maroon" title="c13">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E21:33_1_1"><i><font color="Green" title="E48">A319</font></i></a></i>;<br/><a NAME="E7:33_1_1_11_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_11_1_1"><i><font color="Green" title="E53">A347</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_11_1_1"><i><font color="Green" title="E54">A348</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_11_1_1"><i><font color="Green" title="E55">A349</font></i></a>, <a class="ref" href="finseq_5.html#T25" title="FINSEQ_5:th.25">FINSEQ_5:25</a></i>;<br/><b>hence </b><a NAME="E8:33_1_1_11_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_11"><i><font color="Green" title="E51">A345</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_11_1_1"><i><font color="Green" title="E53">A347</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_11_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_11_1_2"/><i><font color="Green" title="E53">A350</font></i>: 
<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#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:33_1_1_11_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_11_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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_11_1_2"><i><font color="Green" title="E53">A350</font></i></a>, <a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E24:33_1_1"/><i><font color="Green" title="E51">A351</font></i>: 
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>5</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="jordan9.html#E22:33_1_1"><i><font color="Green" title="E49">A342</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1"><i><font color="Green" title="E50">A344</font></i></a>);<br/></i>
<a NAME="E25:33_1_1"/><i><font color="Green" title="E52">A352</font></i>: 
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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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="33_1_1_12"><b>proof </b></a><div class="add">

<b>let </b><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> ;<br/>

<b>assume </b><a NAME="E1:33_1_1_12"/><i><font color="Green" title="E53">A353</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/>

<a NAME="E2:33_1_1_12"/><i><font color="Green" title="E54">A354</font></i>: 
<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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a></i>;<br/>
<a NAME="E3:33_1_1_12"/><i><font color="Green" title="E55">A355</font></i>: 
 <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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a></i>;<br/>
<a NAME="E4:33_1_1_12"/><b>then </b><i><font color="Green" title="E56">A356</font></i>: 
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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_12"><i><font color="Green" title="E53">A353</font></i></a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a></i>;<br/>
<a NAME="E5:33_1_1_12"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_12"><i><font color="Green" title="E53">A353</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_12"><i><font color="Green" title="E55">A355</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/>
<a NAME="E6:33_1_1_12"/><b>then </b>
 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="finseq_4.html#K4" title="FINSEQ_4: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"><font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) ) )
 
<b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_12"><i><font color="Green" title="E54">A354</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_12"><i><font color="Green" title="E56">A356</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/>
<b>hence </b><a NAME="E7:33_1_1_12"/>
( (  <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_12"><i><font color="Green" title="E53">A353</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_12"><i><font color="Green" title="E55">A355</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><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> ] <b>means </b><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="E26:33_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> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1"><i><font color="Green" title="E34">A79</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E27:33_1_1"/><b>then </b><i><font color="Green" title="E53">A357</font></i>: 
<font color="Maroon">S<sub>6</sub></font>[0]
 
<b>by </b><i><a class="ref" href="sppol_2.html#T27" title="SPPOL_2:th.27">SPPOL_2:27</a></i>;<br/>
<a NAME="E28:33_1_1"/><i><font color="Green" title="E54">A358</font></i>: 
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#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_13"><b>proof </b></a><div class="add">

<b>let </b><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> ;<br/>

<b>assume </b><a NAME="E1:33_1_1_13"/><i><font color="Green" title="E55">A359</font></i>: 
<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>
 ;<br/>

<a NAME="E2:33_1_1_13"/><i><font color="Green" title="E56">A360</font></i>: 
<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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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="33_1_1_13_1_1"><b>suppose </b></a><a NAME="E1:33_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
 ;<br/><div class="add"><a NAME="E2:33_1_1_13_1_1"/><b>then </b>
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E3:33_1_1_13_1_1"/><b>then </b>
 <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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_13_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="ref" href="sppol_2.html#T27" title="SPPOL_2:th.27">SPPOL_2:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2"/><i><font color="Green" title="E57">A361</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c11">m</font> = <font color="Maroon" title="c10">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><a NAME="E2:33_1_1_13_1_2"/><i><font color="Green" title="E58">A362</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_13_1_2"><i><font color="Green" title="E57">A361</font></i></a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a></i>;<br/><a NAME="E3:33_1_1_13_1_2"/><i><font color="Green" title="E59">A363</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_13_1_2"><i><font color="Green" title="E57">A361</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><a NAME="E4:33_1_1_13_1_2"/><i><font color="Green" title="E60">A364</font></i>: 
 <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>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a></i>;<br/><b>then consider </b><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> <b> such that </b><br/><a NAME="E6:33_1_1_13_1_2"/><i><font color="Green" title="E61">A365</font></i>: 
( <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> &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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>and </b><br/><a NAME="E7:33_1_1_13_1_2"/><i><font color="Green" title="E62">A366</font></i>: 
( <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> &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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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> )
 <b>and </b><br/><a NAME="E8:33_1_1_13_1_2"/><i><font color="Green" title="E63">A367</font></i>: 
( ( <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_13"><i><font color="Green" title="E56">A360</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_13_1_2"><i><font color="Green" title="E58">A362</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_13_1_2"><i><font color="Green" title="E59">A363</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E9:33_1_1_13_1_2"/><i><font color="Green" title="E64">A368</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i1</font> &amp; <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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j1</font> &amp; <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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1_13_1_2"><i><font color="Green" title="E61">A365</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E10:33_1_1_13_1_2"/><i><font color="Green" title="E65">A369</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i2</font> &amp; <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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j2</font> &amp; <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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_13_1_2"><i><font color="Green" title="E62">A366</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E11:33_1_1_13_1_2"/><b>then </b><i><font color="Green" title="E66">A370</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font>
 <b>by </b><i><a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><a NAME="E12:33_1_1_13_1_2"/><i><font color="Green" title="E67">A371</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E10:33_1_1_13_1_2"><i><font color="Green" title="E65">A369</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><a NAME="E13:33_1_1_13_1_2"/><i><font color="Green" title="E68">A372</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">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>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_13_1_2"><i><font color="Green" title="E58">A362</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_13_1_2"><i><font color="Green" title="E59">A363</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_13_1_2"><i><font color="Green" title="E60">A364</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_13_1_2"><i><font color="Green" title="E61">A365</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_13_1_2"><i><font color="Green" title="E62">A366</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_13_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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="33_1_1_13_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_1"/><i><font color="Green" title="E69">A373</font></i>: 
(  <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E8:33_1_1_13_1_2"><i><font color="Green" title="E63">A367</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_1_1_1_1"/><i><font color="Green" title="E70">A374</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_13_1_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E71">A375</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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> &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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2"><i><font color="Green" title="E57">A361</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_13_1_2"><i><font color="Green" title="E60">A364</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_13_1_2"><i><font color="Green" title="E61">A365</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_13_1_2"><i><font color="Green" title="E62">A366</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2_1_1_1"><i><font color="Green" title="E69">A373</font></i></a></i>;<br/><a NAME="E3:33_1_1_13_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E4:33_1_1_13_1_2_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">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="binarith.html#K3" title="BINARITH:func.3">-'</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="topreal1.html#K3" title="TOPREAL1:func.3">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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_13_1_2"><i><font color="Green" title="E62">A366</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_13_1_2"><i><font color="Green" title="E64">A368</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_13_1_2"><i><font color="Green" title="E65">A369</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_13_1_2"><i><font color="Green" title="E68">A372</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2_1_1_1_1_1_1"><i><font color="Green" title="E70">A374</font></i></a>, <a class="ref" href="goboard7.html#T17" title="GOBOARD7:th.17">GOBOARD7:17</a></i>;<br/><b>hence </b><a NAME="E5:33_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_13"><i><font color="Green" title="E55">A359</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_13_1_2"><i><font color="Green" title="E59">A363</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_13_1_2"><i><font color="Green" title="E60">A364</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_13_1_2_1_1_1_1_1_1"><i><font color="Green" title="E71">A375</font></i></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_1_1_1_2"/><i><font color="Green" title="E70">A376</font></i>: 
( <font color="Maroon" title="c12">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_13_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E71">A377</font></i>: 
( <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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2"><i><font color="Green" title="E57">A361</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_13_1_2"><i><font color="Green" title="E60">A364</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_13_1_2"><i><font color="Green" title="E61">A365</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_13_1_2"><i><font color="Green" title="E62">A366</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2_1_1_1"><i><font color="Green" title="E69">A373</font></i></a></i>;<br/><a NAME="E3:33_1_1_13_1_2_1_1_1_1_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</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="c10">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="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="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>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">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="c14">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</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_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="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> )
 <b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E4:33_1_1_13_1_2_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">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="binarith.html#K3" title="BINARITH:func.3">-'</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="topreal1.html#K3" title="TOPREAL1:func.3">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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_13_1_2"><i><font color="Green" title="E62">A366</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_13_1_2"><i><font color="Green" title="E64">A368</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_13_1_2"><i><font color="Green" title="E65">A369</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_13_1_2"><i><font color="Green" title="E67">A371</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_13_1_2"><i><font color="Green" title="E68">A372</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2_1_1_1_1_1_2"><i><font color="Green" title="E70">A376</font></i></a>, <a class="ref" href="goboard7.html#T18" title="GOBOARD7:th.18">GOBOARD7:18</a></i>;<br/><b>hence </b><a NAME="E5:33_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_13"><i><font color="Green" title="E55">A359</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_13_1_2"><i><font color="Green" title="E59">A363</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_13_1_2"><i><font color="Green" title="E60">A364</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_13_1_2_1_1_1_1_1_2"><i><font color="Green" title="E71">A377</font></i></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_1_1_1_3"/><i><font color="Green" title="E70">A378</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_13_1_2_1_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E71">A379</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2"><i><font color="Green" title="E57">A361</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_13_1_2"><i><font color="Green" title="E60">A364</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_13_1_2"><i><font color="Green" title="E61">A365</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_13_1_2"><i><font color="Green" title="E62">A366</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2_1_1_1"><i><font color="Green" title="E69">A373</font></i></a></i>;<br/><a NAME="E3:33_1_1_13_1_2_1_1_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E4:33_1_1_13_1_2_1_1_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">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="binarith.html#K3" title="BINARITH:func.3">-'</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="topreal1.html#K3" title="TOPREAL1:func.3">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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_13_1_2"><i><font color="Green" title="E62">A366</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_13_1_2"><i><font color="Green" title="E64">A368</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_13_1_2"><i><font color="Green" title="E65">A369</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_13_1_2"><i><font color="Green" title="E68">A372</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2_1_1_1_1_1_3"><i><font color="Green" title="E70">A378</font></i></a>, <a class="ref" href="goboard7.html#T19" title="GOBOARD7:th.19">GOBOARD7:19</a></i>;<br/><b>hence </b><a NAME="E5:33_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_13"><i><font color="Green" title="E55">A359</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_13_1_2"><i><font color="Green" title="E59">A363</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_13_1_2"><i><font color="Green" title="E60">A364</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_13_1_2_1_1_1_1_1_3"><i><font color="Green" title="E71">A379</font></i></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_1_1_1_4"/><i><font color="Green" title="E70">A380</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_13_1_2_1_1_1_1_1_4"/><b>then </b><i><font color="Green" title="E71">A381</font></i>: 
( <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="binarith.html#K3" title="BINARITH:func.3">-'</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> &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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2"><i><font color="Green" title="E57">A361</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_13_1_2"><i><font color="Green" title="E60">A364</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_13_1_2"><i><font color="Green" title="E61">A365</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_13_1_2"><i><font color="Green" title="E62">A366</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2_1_1_1"><i><font color="Green" title="E69">A373</font></i></a></i>;<br/><a NAME="E3:33_1_1_13_1_2_1_1_1_1_1_4"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E4:33_1_1_13_1_2_1_1_1_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">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="binarith.html#K3" title="BINARITH:func.3">-'</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="topreal1.html#K3" title="TOPREAL1:func.3">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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_13_1_2"><i><font color="Green" title="E62">A366</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_13_1_2"><i><font color="Green" title="E64">A368</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_13_1_2"><i><font color="Green" title="E65">A369</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_13_1_2"><i><font color="Green" title="E66">A370</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_13_1_2"><i><font color="Green" title="E68">A372</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2_1_1_1_1_1_4"><i><font color="Green" title="E70">A380</font></i></a>, <a class="ref" href="goboard7.html#T20" title="GOBOARD7:th.20">GOBOARD7:20</a></i>;<br/><b>hence </b><a NAME="E5:33_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_13"><i><font color="Green" title="E55">A359</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_13_1_2"><i><font color="Green" title="E59">A363</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_13_1_2"><i><font color="Green" title="E60">A364</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_13_1_2_1_1_1_1_1_4"><i><font color="Green" title="E71">A381</font></i></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_13_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_2"/><i><font color="Green" title="E69">A382</font></i>: 
(  <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E8:33_1_1_13_1_2"><i><font color="Green" title="E63">A367</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E70">A383</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_13_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E71">A384</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2"><i><font color="Green" title="E57">A361</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_13_1_2"><i><font color="Green" title="E60">A364</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_13_1_2"><i><font color="Green" title="E61">A365</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_13_1_2"><i><font color="Green" title="E62">A366</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2_1_1_2"><i><font color="Green" title="E69">A382</font></i></a></i>;<br/><a NAME="E3:33_1_1_13_1_2_1_1_2_1_1_1"/><b>then </b>
( 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#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> &amp; <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E70">A383</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E4:33_1_1_13_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">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="binarith.html#K3" title="BINARITH:func.3">-'</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="topreal1.html#K3" title="TOPREAL1:func.3">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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_13_1_2"><i><font color="Green" title="E62">A366</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_13_1_2"><i><font color="Green" title="E64">A368</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_13_1_2"><i><font color="Green" title="E68">A372</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E70">A383</font></i></a>, <a class="ref" href="goboard7.html#T15" title="GOBOARD7:th.15">GOBOARD7:15</a></i>;<br/><b>hence </b><a NAME="E5:33_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_13"><i><font color="Green" title="E55">A359</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_13_1_2"><i><font color="Green" title="E59">A363</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_13_1_2"><i><font color="Green" title="E60">A364</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E71">A384</font></i></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E70">A385</font></i>: 
( <font color="Maroon" title="c12">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_13_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E71">A386</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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> &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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2"><i><font color="Green" title="E57">A361</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_13_1_2"><i><font color="Green" title="E60">A364</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_13_1_2"><i><font color="Green" title="E61">A365</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_13_1_2"><i><font color="Green" title="E62">A366</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2_1_1_2"><i><font color="Green" title="E69">A382</font></i></a></i>;<br/><a NAME="E3:33_1_1_13_1_2_1_1_2_1_1_2"/><b>then </b>
( 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#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_13_1_2_1_1_2_1_1_2"><i><font color="Green" title="E70">A385</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E4:33_1_1_13_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">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="binarith.html#K3" title="BINARITH:func.3">-'</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="topreal1.html#K3" title="TOPREAL1:func.3">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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_13_1_2"><i><font color="Green" title="E62">A366</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_13_1_2"><i><font color="Green" title="E64">A368</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_13_1_2"><i><font color="Green" title="E68">A372</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2_1_1_2_1_1_2"><i><font color="Green" title="E70">A385</font></i></a>, <a class="ref" href="goboard7.html#T16" title="GOBOARD7:th.16">GOBOARD7:16</a></i>;<br/><b>hence </b><a NAME="E5:33_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_13"><i><font color="Green" title="E55">A359</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_13_1_2"><i><font color="Green" title="E59">A363</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_13_1_2"><i><font color="Green" title="E60">A364</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_13_1_2_1_1_2_1_1_2"><i><font color="Green" title="E71">A386</font></i></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_2_1_1_3"/><i><font color="Green" title="E70">A387</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_13_1_2_1_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E71">A388</font></i>: 
( <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="binarith.html#K3" title="BINARITH:func.3">-'</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> &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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2"><i><font color="Green" title="E57">A361</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_13_1_2"><i><font color="Green" title="E60">A364</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_13_1_2"><i><font color="Green" title="E61">A365</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_13_1_2"><i><font color="Green" title="E62">A366</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2_1_1_2"><i><font color="Green" title="E69">A382</font></i></a></i>;<br/><a NAME="E3:33_1_1_13_1_2_1_1_2_1_1_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E4:33_1_1_13_1_2_1_1_2_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">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="binarith.html#K3" title="BINARITH:func.3">-'</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="topreal1.html#K3" title="TOPREAL1:func.3">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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_13_1_2"><i><font color="Green" title="E62">A366</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_13_1_2"><i><font color="Green" title="E64">A368</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_13_1_2"><i><font color="Green" title="E66">A370</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_13_1_2"><i><font color="Green" title="E68">A372</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2_1_1_2_1_1_3"><i><font color="Green" title="E70">A387</font></i></a>, <a class="ref" href="goboard7.html#T16" title="GOBOARD7:th.16">GOBOARD7:16</a></i>;<br/><b>hence </b><a NAME="E5:33_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_13"><i><font color="Green" title="E55">A359</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_13_1_2"><i><font color="Green" title="E59">A363</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_13_1_2"><i><font color="Green" title="E60">A364</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_13_1_2_1_1_2_1_1_3"><i><font color="Green" title="E71">A388</font></i></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_2_1_1_4"/><i><font color="Green" title="E70">A389</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_13_1_2_1_1_2_1_1_4"/><b>then </b><i><font color="Green" title="E71">A390</font></i>: 
( <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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2"><i><font color="Green" title="E57">A361</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_13_1_2"><i><font color="Green" title="E60">A364</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_13_1_2"><i><font color="Green" title="E61">A365</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_13_1_2"><i><font color="Green" title="E62">A366</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2_1_1_2"><i><font color="Green" title="E69">A382</font></i></a></i>;<br/><a NAME="E3:33_1_1_13_1_2_1_1_2_1_1_4"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E4:33_1_1_13_1_2_1_1_2_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">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="binarith.html#K3" title="BINARITH:func.3">-'</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="topreal1.html#K3" title="TOPREAL1:func.3">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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_13_1_2"><i><font color="Green" title="E62">A366</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_13_1_2"><i><font color="Green" title="E64">A368</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_13_1_2"><i><font color="Green" title="E67">A371</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_13_1_2"><i><font color="Green" title="E68">A372</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2_1_1_2_1_1_4"><i><font color="Green" title="E70">A389</font></i></a>, <a class="ref" href="goboard7.html#T15" title="GOBOARD7:th.15">GOBOARD7:15</a></i>;<br/><b>hence </b><a NAME="E5:33_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_13"><i><font color="Green" title="E55">A359</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_13_1_2"><i><font color="Green" title="E59">A363</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_13_1_2"><i><font color="Green" title="E60">A364</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_13_1_2_1_1_2_1_1_4"><i><font color="Green" title="E71">A390</font></i></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_13_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_3"/><i><font color="Green" title="E69">A391</font></i>: 
 <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E8:33_1_1_13_1_2"><i><font color="Green" title="E63">A367</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_3_1_1_1"/><i><font color="Green" title="E70">A392</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_13_1_2_1_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E71">A393</font></i>: 
( <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="binarith.html#K3" title="BINARITH:func.3">-'</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> &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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2"><i><font color="Green" title="E57">A361</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_13_1_2"><i><font color="Green" title="E60">A364</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_13_1_2"><i><font color="Green" title="E61">A365</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_13_1_2"><i><font color="Green" title="E62">A366</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2_1_1_3"><i><font color="Green" title="E69">A391</font></i></a></i>;<br/><a NAME="E3:33_1_1_13_1_2_1_1_3_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</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="c10">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="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="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="c14">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">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> <span class="p2">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span>)</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_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="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> )
 <b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E4:33_1_1_13_1_2_1_1_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">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="binarith.html#K3" title="BINARITH:func.3">-'</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="topreal1.html#K3" title="TOPREAL1:func.3">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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_13_1_2"><i><font color="Green" title="E62">A366</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_13_1_2"><i><font color="Green" title="E64">A368</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_13_1_2"><i><font color="Green" title="E65">A369</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_13_1_2"><i><font color="Green" title="E66">A370</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_13_1_2"><i><font color="Green" title="E68">A372</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2_1_1_3_1_1_1"><i><font color="Green" title="E70">A392</font></i></a>, <a class="ref" href="goboard7.html#T18" title="GOBOARD7:th.18">GOBOARD7:18</a></i>;<br/><b>hence </b><a NAME="E5:33_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_13"><i><font color="Green" title="E55">A359</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_13_1_2"><i><font color="Green" title="E59">A363</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_13_1_2"><i><font color="Green" title="E60">A364</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_13_1_2_1_1_3_1_1_1"><i><font color="Green" title="E71">A393</font></i></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_3_1_1_2"/><i><font color="Green" title="E70">A394</font></i>: 
( <font color="Maroon" title="c12">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_13_1_2_1_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E71">A395</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2"><i><font color="Green" title="E57">A361</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_13_1_2"><i><font color="Green" title="E60">A364</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_13_1_2"><i><font color="Green" title="E61">A365</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_13_1_2"><i><font color="Green" title="E62">A366</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2_1_1_3"><i><font color="Green" title="E69">A391</font></i></a></i>;<br/><a NAME="E3:33_1_1_13_1_2_1_1_3_1_1_2"/><b>then </b>
( 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#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c15">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E4:33_1_1_13_1_2_1_1_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">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="binarith.html#K3" title="BINARITH:func.3">-'</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="topreal1.html#K3" title="TOPREAL1:func.3">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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_13_1_2"><i><font color="Green" title="E62">A366</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_13_1_2"><i><font color="Green" title="E64">A368</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_13_1_2"><i><font color="Green" title="E65">A369</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_13_1_2"><i><font color="Green" title="E68">A372</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2_1_1_3_1_1_2"><i><font color="Green" title="E70">A394</font></i></a>, <a class="ref" href="goboard7.html#T20" title="GOBOARD7:th.20">GOBOARD7:20</a></i>;<br/><b>hence </b><a NAME="E5:33_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_13"><i><font color="Green" title="E55">A359</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_13_1_2"><i><font color="Green" title="E59">A363</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_13_1_2"><i><font color="Green" title="E60">A364</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_13_1_2_1_1_3_1_1_2"><i><font color="Green" title="E71">A395</font></i></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_3_1_1_3"/><i><font color="Green" title="E70">A396</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_13_1_2_1_1_3_1_1_3"/><b>then </b><i><font color="Green" title="E71">A397</font></i>: 
( <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="binarith.html#K3" title="BINARITH:func.3">-'</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#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2"><i><font color="Green" title="E57">A361</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_13_1_2"><i><font color="Green" title="E60">A364</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_13_1_2"><i><font color="Green" title="E61">A365</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_13_1_2"><i><font color="Green" title="E62">A366</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2_1_1_3"><i><font color="Green" title="E69">A391</font></i></a></i>;<br/><a NAME="E3:33_1_1_13_1_2_1_1_3_1_1_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</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="c10">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="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="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>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">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="c14">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</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_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="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> )
 <b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E4:33_1_1_13_1_2_1_1_3_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">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="binarith.html#K3" title="BINARITH:func.3">-'</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="topreal1.html#K3" title="TOPREAL1:func.3">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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="binarith.html#K3" title="BINARITH:func.3">-'</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_13_1_2"><i><font color="Green" title="E62">A366</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_13_1_2"><i><font color="Green" title="E64">A368</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_13_1_2"><i><font color="Green" title="E65">A369</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_13_1_2"><i><font color="Green" title="E67">A371</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_13_1_2"><i><font color="Green" title="E68">A372</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2_1_1_3_1_1_3"><i><font color="Green" title="E70">A396</font></i></a>, <a class="ref" href="goboard7.html#T17" title="GOBOARD7:th.17">GOBOARD7:17</a></i>;<br/><b>hence </b><a NAME="E5:33_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_13"><i><font color="Green" title="E55">A359</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_13_1_2"><i><font color="Green" title="E59">A363</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_13_1_2"><i><font color="Green" title="E60">A364</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_13_1_2_1_1_3_1_1_3"><i><font color="Green" title="E71">A397</font></i></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_3_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_3_1_1_4"/><i><font color="Green" title="E70">A398</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_13_1_2_1_1_3_1_1_4"/><b>then </b><i><font color="Green" title="E71">A399</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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> &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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2"><i><font color="Green" title="E57">A361</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_13_1_2"><i><font color="Green" title="E60">A364</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_13_1_2"><i><font color="Green" title="E61">A365</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_13_1_2"><i><font color="Green" title="E62">A366</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2_1_1_3"><i><font color="Green" title="E69">A391</font></i></a></i>;<br/><a NAME="E3:33_1_1_13_1_2_1_1_3_1_1_4"/><b>then </b>
( 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#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</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="c10">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="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="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="c14">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">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> <span class="p2">(<span class="default"><font color="Maroon" title="c14">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c15">j2</font></span>)</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_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="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> )
 <b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E4:33_1_1_13_1_2_1_1_3_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">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="binarith.html#K3" title="BINARITH:func.3">-'</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="topreal1.html#K3" title="TOPREAL1:func.3">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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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#K1" title="NAT_1:func.1">+</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_13_1_2"><i><font color="Green" title="E62">A366</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_13_1_2"><i><font color="Green" title="E64">A368</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_13_1_2"><i><font color="Green" title="E65">A369</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_13_1_2"><i><font color="Green" title="E68">A372</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_13_1_2_1_1_3_1_1_4"><i><font color="Green" title="E70">A398</font></i></a>, <a class="ref" href="goboard7.html#T19" title="GOBOARD7:th.19">GOBOARD7:19</a></i>;<br/><b>hence </b><a NAME="E5:33_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_13"><i><font color="Green" title="E55">A359</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_13_1_2"><i><font color="Green" title="E59">A363</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_13_1_2"><i><font color="Green" title="E60">A364</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_13_1_2_1_1_3_1_1_4"><i><font color="Green" title="E71">A399</font></i></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_13_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E15:33_1_1_13_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E29:33_1_1"/><i><font color="Green" title="E55">A400</font></i>: 
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>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="jordan9.html#E27:33_1_1"><i><font color="Green" title="E53">A357</font></i></a>, <a class="txt" href="jordan9.html#E28:33_1_1"><i><font color="Green" title="E54">A358</font></i></a>);<br/></i>
<b>defpred </b><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> ] <b>means </b>( $1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  ex <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 <br/>( <font color="Olive" title="b1">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> $1</span>)</span> &amp; <font color="Olive" title="b1">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> $1</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> $1</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">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> $1</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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> $1</span>)</span></span>)</span> ) );<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_14"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_14"/><i><font color="Green" title="E56">A401</font></i>: 
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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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>
 ;<br/><b>defpred </b><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> ] <b>means </b><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/><a NAME="E2:33_1_1_14"/><i><font color="Green" title="E57">A402</font></i>: 
<font color="Maroon">S<sub>8</sub></font>[0]
 <b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1"><i><font color="Green" title="E34">A79</font></i></a></i>;<br/><a NAME="E3:33_1_1_14"/><i><font color="Green" title="E58">A403</font></i>: 
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#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_14_1"><b>proof </b></a><div class="add">

<b>let </b><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> ;<br/>

<b>assume </b><a NAME="E1:33_1_1_14_1"/><i><font color="Green" title="E59">A404</font></i>: 
<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>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_14_1_1"><b>now </b></a><div class="add"><b>let </b><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> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_14_1_1"/><i><font color="Green" title="E60">A405</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E2:33_1_1_14_1_1"/><i><font color="Green" title="E61">A406</font></i>: 
<span class="p1">(<span class="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">m</font>
 ;<br/><b>consider </b><font color="Maroon" title="c13">i</font>, <font color="Maroon" title="c14">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> <b> such that </b><br/><a NAME="E4:33_1_1_14_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">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>
 <b>and </b><br/><a NAME="E5:33_1_1_14_1_1"/><i><font color="Green" title="E62">A407</font></i>: 
<font 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#K1" title="NAT_1:func.1">+</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">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E21:33_1_1"><i><font color="Green" title="E48">A319</font></i></a></i>;<br/><a NAME="E6:33_1_1_14_1_1"/><i><font color="Green" title="E63">A408</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> &amp; <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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp; 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="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_14_1_1"><i><font color="Green" title="E60">A405</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:33_1_1_14_1_1"/><i><font color="Green" title="E64">A409</font></i>: 
(  <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;  <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a></i>;<br/><a NAME="E8:33_1_1_14_1_1"/><i><font color="Green" title="E65">A410</font></i>: 
1 <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#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_14_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1_14_1_1"><i><font color="Green" title="E63">A408</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_14_1_1"><i><font color="Green" title="E64">A409</font></i></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_14_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_14_1_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> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_14_1_1_1_1"/><b>then </b><i><font color="Green" title="E66">A411</font></i>: 
( <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> &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> <font color="Maroon" title="c10">k</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1_14_1_1"><i><font color="Green" title="E63">A408</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_14_1_1"><i><font color="Green" title="E64">A409</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:33_1_1_14_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">n</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">m</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1_14_1_1"><i><font color="Green" title="E62">A407</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><b>hence </b><a NAME="E4:33_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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_14_1"><i><font color="Green" title="E59">A404</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_14_1_1"><i><font color="Green" title="E61">A406</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_14_1_1_1_1"><i><font color="Green" title="E66">A411</font></i></a>, <a class="ref" href="partfun2.html#T17" title="PARTFUN2:th.17">PARTFUN2:17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_14_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_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#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_14"><i><font color="Green" title="E56">A401</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_14_1_1"><i><font color="Green" title="E60">A405</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_14_1_1"><i><font color="Green" title="E61">A406</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_14_1_1"><i><font color="Green" title="E64">A409</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_14_1_1"><i><font color="Green" title="E65">A410</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_14_1_1_1_3"><b>suppose </b></a><a NAME="E1:33_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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_14"><i><font color="Green" title="E56">A401</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_14_1_1"><i><font color="Green" title="E60">A405</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_14_1_1"><i><font color="Green" title="E61">A406</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_14_1_1"><i><font color="Green" title="E64">A409</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_14_1_1"><i><font color="Green" title="E65">A410</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_14_1_1_1_4"><b>suppose </b></a><a NAME="E1:33_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_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>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:33_1_1_14_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="partfun2.html#T16" title="PARTFUN2:th.16">PARTFUN2:16</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:33_1_1_14"/><i><font color="Green" title="E59">A412</font></i>: 
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>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="jordan9.html#E2:33_1_1_14"><i><font color="Green" title="E57">A402</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_14"><i><font color="Green" title="E58">A403</font></i></a>);<br/></i><a NAME="E5:33_1_1_14"/><i><font color="Green" title="E60">A413</font></i>: 
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#K4" title="CARD_1:func.4">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="33_1_1_14_2"><b>proof </b></a><div class="add">

<b>let </b><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> ;<br/>

<a NAME="E1:33_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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_14"><i><font color="Green" title="E59">A412</font></i></a></i>;<br/>
<b>hence </b> <a href="card_1.html#K4" title="CARD_1:func.4">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>
<b>by </b><i><a class="ref" href="finseq_4.html#T77" title="FINSEQ_4:th.77">FINSEQ_4:77</a></i>
<br/>.= 
<font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div><b>set </b><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#K2" title="NAT_1:func.2">*</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#K1" title="NAT_1:func.1">+</a> 1;<br/><a NAME="E6:33_1_1_14"/><i><font color="Green" title="E61">A414</font></i>: 
<span class="p1">(<span class="default"><span 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#K2" title="NAT_1:func.2">*</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#K1" title="NAT_1:func.1">+</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#K2" title="NAT_1:func.2">*</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>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E7:33_1_1_14"/><i><font color="Green" title="E62">A415</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">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#K2" title="NAT_1:func.2">*</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>
 <b>by </b><i><a class="ref" href="gobrd13.html#T8" title="GOBRD13:th.8">GOBRD13:8</a></i>;<br/><a NAME="E8:33_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#K2" title="NAT_1:func.2">*</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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a></i>;<br/><a NAME="E9:33_1_1_14"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</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"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p5">(<span class="default"><a href="jordan8.html#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> <span class="p4">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p5">(<span class="default"><a href="jordan8.html#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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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>
 <b>by </b><i><a class="ref" href="gobrd13.html#T9" title="GOBRD13:th.9">GOBRD13:9</a></i>;<br/><a NAME="E10:33_1_1_14"/><b>then </b>
 <a href="card_1.html#K4" title="CARD_1:func.4">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#K2" title="NAT_1:func.2">*</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#K1" title="NAT_1:func.1">+</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#K4" title="CARD_1:func.4">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>
 <b>by </b><i><a class="ref" href="card_1.html#T80" title="CARD_1:th.80">CARD_1:80</a></i>;<br/><a NAME="E11:33_1_1_14"/><b>then </b>
 <a href="card_1.html#K4" title="CARD_1:func.4">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#K2" title="NAT_1:func.2">*</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#K1" title="NAT_1:func.1">+</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#K2" title="NAT_1:func.2">*</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_14"><i><font color="Green" title="E62">A415</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E12:33_1_1_14"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1_14"><i><font color="Green" title="E60">A413</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_14"><i><font color="Green" title="E61">A414</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E31:33_1_1"/><b>then </b><i><font color="Green" title="E56">A416</font></i>: 
 ex <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>]
 
;<br/>
<b>consider </b><font color="Maroon" title="c10">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> <b> such that </b><br/><a NAME="E33:33_1_1"/><i><font color="Green" title="E57">A417</font></i>: 
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c10">k</font>]
 <b>and </b><br/><a NAME="E34:33_1_1"/><i><font color="Green" title="E58">A418</font></i>: 
for <font color="Olive" title="b1">l</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">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>
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="jordan9.html#E31:33_1_1"><i><font color="Green" title="E56">A416</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon" title="c11">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> <b> such that </b><br/><a NAME="E36:33_1_1"/><i><font color="Green" title="E59">A419</font></i>: 
<font color="Maroon" title="c11">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>
 <b>and </b><br/><a NAME="E37:33_1_1"/><i><font color="Green" title="E60">A420</font></i>: 
<font color="Maroon" title="c11">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="c10">k</font></span>)</span>
 <b>and </b><br/><a NAME="E38:33_1_1"/><i><font color="Green" title="E61">A421</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="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_4.html#K4" title="FINSEQ_4:func.4">/.</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_4.html#K4" title="FINSEQ_4:func.4">/.</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E33:33_1_1"><i><font color="Green" title="E57">A417</font></i></a></i>;<br/>
<a NAME="E39:33_1_1"/><i><font color="Green" title="E62">A422</font></i>: 
 <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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">f</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> 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> <b>by </b><i><a class="txt" href="jordan9.html#E33:33_1_1"><i><font color="Green" title="E57">A417</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E41:33_1_1"/><i><font color="Green" title="E63">A423</font></i>: 
<font color="Maroon" title="c12">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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a></i>;<br/>
<a NAME="E42:33_1_1"/><i><font color="Green" title="E64">A424</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font> &amp; <font color="Maroon" title="c11">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="c12">f</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E36:33_1_1"><i><font color="Green" title="E59">A419</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E43:33_1_1"/><b>then </b><i><font color="Green" title="E65">A425</font></i>: 
<font color="Maroon" title="c11">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="c12">f</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E37:33_1_1"><i><font color="Green" title="E60">A420</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E44:33_1_1"/><b>then </b>
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="c12">f</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E45:33_1_1"/><b>then </b><i><font color="Green" title="E66">A426</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">f</font> = <font color="Maroon" title="c12">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> <b>by </b><i><a class="txt" href="jordan9.html#E29:33_1_1"><i><font color="Green" title="E55">A400</font></i></a>, <a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></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></i>;<br/>
<b>set </b><font color="Maroon" title="c14">g</font> = <font color="Maroon" title="c13">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>;<br/>
<a NAME="E47:33_1_1"/><i><font color="Green" title="E67">A427</font></i>: 
<font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c13">f</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E43:33_1_1"><i><font color="Green" title="E65">A425</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E48:33_1_1"/>
( <font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font> &amp; <font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E49:33_1_1"/><b>then </b>
<font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E50:33_1_1"/><b>then </b><i><font color="Green" title="E68">A428</font></i>: 
<font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c13">f</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E47:33_1_1"><i><font color="Green" title="E67">A427</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E51:33_1_1"/><b>then </b><i><font color="Green" title="E69">A429</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c13">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E52:33_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c13">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E50:33_1_1"><i><font color="Green" title="E68">A428</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">g</font> = <font color="Maroon" title="c13">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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> <b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E54:33_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="c15">g</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<b>then </b><i><font color="Green" title="E70">A430</font></i>: <font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>
<br/>.= 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">m</font>
<b>by </b><i><a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>
;<br/>
<a NAME="E56:33_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">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="c15">g</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<b>then </b><i><font color="Green" title="E71">A431</font></i>: <font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> = 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>
<br/>.= 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">f</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E51:33_1_1"><i><font color="Green" title="E69">A429</font></i></a></i>
;<br/>
<a NAME="E58:33_1_1"/><i><font color="Green" title="E72">A432</font></i>: 
<font color="Maroon" title="c15">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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="ref" href="jordan8.html#T5" title="JORDAN8:th.5">JORDAN8:5</a></i>;<br/>
<a NAME="E59:33_1_1"/><b>then </b><i><font color="Green" title="E73">A433</font></i>: 
<font color="Maroon" title="c15">g</font> is <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a>
 
<b>by </b><i><a class="ref" href="jordan8.html#T7" title="JORDAN8:th.7">JORDAN8:7</a></i>;<br/>
<a NAME="E60:33_1_1"/><i><font color="Green" title="E74">A434</font></i>: 
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="c15">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="c15">g</font> &amp; <font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="33_1_1_17"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">j</font>, <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> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_17"/><i><font color="Green" title="E75">A435</font></i>: 
( 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="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="c15">g</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_17"/><i><font color="Green" title="E76">A436</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j</font> &amp; <font color="Maroon" title="c16">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="c15">g</font> )
 <b>and </b><br/><a NAME="E3:33_1_1_17"/><i><font color="Green" title="E77">A437</font></i>: 
<font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">j</font>
 <b>and </b><br/><a NAME="E4:33_1_1_17"/><i><font color="Green" title="E78">A438</font></i>: 
<font color="Maroon" title="c17">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">j</font>
 ;<br/>

<a NAME="E5:33_1_1_17"/>
<font color="Maroon" title="c17">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="c15">g</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_17"><i><font color="Green" title="E75">A435</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E6:33_1_1_17"/><b>then </b><i><font color="Green" title="E79">A439</font></i>: 
<font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>;<br/>
<a NAME="E7:33_1_1_17"/>
<font color="Maroon" title="c16">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="c15">g</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_17"><i><font color="Green" title="E76">A436</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:33_1_1_17"/><b>then </b><i><font color="Green" title="E80">A440</font></i>: 
<font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_17_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_17_1"/>
( <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">j</font> or <font color="Maroon" title="c16">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">i</font> )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_17"><i><font color="Green" title="E78">A438</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_17_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_17_1_1"/><i><font color="Green" title="E81">A441</font></i>: 
<font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">j</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c18">l</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">j</font>;<br/><b>set </b><font color="Maroon" title="c19">m'</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">i</font>;<br/><a NAME="E2:33_1_1_17_1_1"/><i><font color="Green" title="E82">A442</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">j</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_17_1_1"><i><font color="Green" title="E81">A441</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E3:33_1_1_17_1_1"/><i><font color="Green" title="E83">A443</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E51:33_1_1"><i><font color="Green" title="E69">A429</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_17"><i><font color="Green" title="E76">A436</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E4:33_1_1_17_1_1"/><b>then </b><i><font color="Green" title="E84">A444</font></i>: 
<font color="Maroon" title="c13">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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E24:33_1_1"><i><font color="Green" title="E51">A351</font></i></a></i>;<br/><a NAME="E5:33_1_1_17_1_1"/>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">j</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E6:33_1_1_17_1_1"/><b>then </b><i><font color="Green" title="E85">A445</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">j</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_17"><i><font color="Green" title="E76">A436</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:33_1_1_17_1_1"/><i><font color="Green" title="E86">A446</font></i>: 
 <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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">j</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a></i>;<br/><a NAME="E8:33_1_1_17_1_1"/>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">i</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E9:33_1_1_17_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">i</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_17"><i><font color="Green" title="E75">A435</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:33_1_1_17_1_1"/><b>then </b><i><font color="Green" title="E87">A447</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_17_1_1"><i><font color="Green" title="E82">A442</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_17_1_1"><i><font color="Green" title="E86">A446</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:33_1_1_17_1_1"/><b>then </b><i><font color="Green" title="E88">A448</font></i>: 
<span class="p1">(<span class="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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">j</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_17_1_1"><i><font color="Green" title="E84">A444</font></i></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></i>;<br/><a NAME="E12:33_1_1_17_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1_17_1_1"><i><font color="Green" title="E85">A445</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_17_1_1"><i><font color="Green" title="E86">A446</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E13:33_1_1_17_1_1"/><b>then </b>
<span class="p1">(<span class="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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">j</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_17_1_1"><i><font color="Green" title="E84">A444</font></i></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></i>;<br/><b>hence </b><a NAME="E14:33_1_1_17_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E34:33_1_1"><i><font color="Green" title="E58">A418</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_17"><i><font color="Green" title="E77">A437</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_17"><i><font color="Green" title="E79">A439</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_17"><i><font color="Green" title="E80">A440</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_17_1_1"><i><font color="Green" title="E82">A442</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_17_1_1"><i><font color="Green" title="E83">A443</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_17_1_1"><i><font color="Green" title="E85">A445</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_17_1_1"><i><font color="Green" title="E86">A446</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_17_1_1"><i><font color="Green" title="E87">A447</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_17_1_1"><i><font color="Green" title="E88">A448</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_17_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_17_1_2"/><i><font color="Green" title="E81">A449</font></i>: 
<font color="Maroon" title="c16">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">i</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c18">l</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">i</font>;<br/><b>set </b><font color="Maroon" title="c19">m'</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">j</font>;<br/><a NAME="E2:33_1_1_17_1_2"/><i><font color="Green" title="E82">A450</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">i</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_17_1_2"><i><font color="Green" title="E81">A449</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E3:33_1_1_17_1_2"/><i><font color="Green" title="E83">A451</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E51:33_1_1"><i><font color="Green" title="E69">A429</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_17"><i><font color="Green" title="E75">A435</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E4:33_1_1_17_1_2"/><b>then </b><i><font color="Green" title="E84">A452</font></i>: 
<font color="Maroon" title="c13">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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E24:33_1_1"><i><font color="Green" title="E51">A351</font></i></a></i>;<br/><a NAME="E5:33_1_1_17_1_2"/>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">i</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E6:33_1_1_17_1_2"/><b>then </b><i><font color="Green" title="E85">A453</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">i</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_17"><i><font color="Green" title="E75">A435</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:33_1_1_17_1_2"/><i><font color="Green" title="E86">A454</font></i>: 
 <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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">i</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a></i>;<br/><a NAME="E8:33_1_1_17_1_2"/>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">j</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E9:33_1_1_17_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">j</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_17"><i><font color="Green" title="E76">A436</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:33_1_1_17_1_2"/><b>then </b><i><font color="Green" title="E87">A455</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_17_1_2"><i><font color="Green" title="E82">A450</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_17_1_2"><i><font color="Green" title="E86">A454</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:33_1_1_17_1_2"/><b>then </b><i><font color="Green" title="E88">A456</font></i>: 
<span class="p1">(<span class="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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">i</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_17_1_2"><i><font color="Green" title="E84">A452</font></i></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></i>;<br/><a NAME="E12:33_1_1_17_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1_17_1_2"><i><font color="Green" title="E85">A453</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_17_1_2"><i><font color="Green" title="E86">A454</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E13:33_1_1_17_1_2"/><b>then </b>
<span class="p1">(<span class="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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">i</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_17_1_2"><i><font color="Green" title="E84">A452</font></i></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></i>;<br/><b>hence </b><a NAME="E14:33_1_1_17_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E34:33_1_1"><i><font color="Green" title="E58">A418</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_17"><i><font color="Green" title="E77">A437</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_17"><i><font color="Green" title="E79">A439</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_17"><i><font color="Green" title="E80">A440</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_17_1_2"><i><font color="Green" title="E82">A450</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_17_1_2"><i><font color="Green" title="E83">A451</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_17_1_2"><i><font color="Green" title="E85">A453</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_17_1_2"><i><font color="Green" title="E86">A454</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_17_1_2"><i><font color="Green" title="E87">A455</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_17_1_2"><i><font color="Green" title="E88">A456</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E61:33_1_1"/><i><font color="Green" title="E75">A457</font></i>: 
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="c15">g</font> holds <br/><font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_18"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">j</font>, <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> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_18"/><i><font color="Green" title="E76">A458</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">i</font>
 <b>and </b><br/><a NAME="E2:33_1_1_18"/><i><font color="Green" title="E77">A459</font></i>: 
<font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">j</font>
 <b>and </b><br/><a NAME="E3:33_1_1_18"/><i><font color="Green" title="E78">A460</font></i>: 
<font color="Maroon" title="c16">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="c15">g</font>
 <b>and </b><br/><a NAME="E4:33_1_1_18"/><i><font color="Green" title="E79">A461</font></i>: 
<font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">j</font>
 ;<br/>

<a NAME="E5:33_1_1_18"/><i><font color="Green" title="E80">A462</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">j</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_18"><i><font color="Green" title="E76">A458</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_18"><i><font color="Green" title="E77">A459</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:33_1_1_18"/><i><font color="Green" title="E81">A463</font></i>: 
<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="c15">g</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_18"><i><font color="Green" title="E77">A459</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_18"><i><font color="Green" title="E78">A460</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:33_1_1_18"/><b>then </b><i><font color="Green" title="E82">A464</font></i>: 
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="c15">g</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_18"><i><font color="Green" title="E76">A458</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_18_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_18_1"/>
( <font color="Maroon" title="c16">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="c15">g</font> or <font color="Maroon" title="c16">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="c15">g</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_18_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_18_1_1"/>
<font color="Maroon" title="c16">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="c15">g</font>
 ;<br/><div class="add"><a NAME="E2:33_1_1_18_1_1"/><b>then </b>
<font color="Maroon" title="c16">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="c15">g</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_18"><i><font color="Green" title="E78">A460</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_18_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_18"><i><font color="Green" title="E76">A458</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_18"><i><font color="Green" title="E77">A459</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_18"><i><font color="Green" title="E79">A461</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_18"><i><font color="Green" title="E80">A462</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_18"><i><font color="Green" title="E81">A463</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_18_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_18_1_2"/>
<font color="Maroon" title="c16">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="c15">g</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_18_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E38:33_1_1"><i><font color="Green" title="E61">A421</font></i></a>, <a class="txt" href="jordan9.html#E55:33_1_1"><i><font color="Green" title="E70">A430</font></i></a>, <a class="txt" href="jordan9.html#E57:33_1_1"><i><font color="Green" title="E71">A431</font></i></a>, <a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_18"><i><font color="Green" title="E76">A458</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_18"><i><font color="Green" title="E77">A459</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_18"><i><font color="Green" title="E79">A461</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_18"><i><font color="Green" title="E82">A464</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E62:33_1_1"/><i><font color="Green" title="E76">A465</font></i>: 
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="c15">g</font> holds <br/><font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_19"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">j</font>, <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> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_19"/><i><font color="Green" title="E77">A466</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i</font>
 <b>and </b><br/><a NAME="E2:33_1_1_19"/><i><font color="Green" title="E78">A467</font></i>: 
<font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">j</font>
 <b>and </b><br/><a NAME="E3:33_1_1_19"/><i><font color="Green" title="E79">A468</font></i>: 
<font color="Maroon" title="c16">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="c15">g</font>
 <b>and </b><br/><a NAME="E4:33_1_1_19"/><i><font color="Green" title="E80">A469</font></i>: 
<font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">j</font>
 ;<br/>

<a NAME="E5:33_1_1_19"/><i><font color="Green" title="E81">A470</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">j</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_19"><i><font color="Green" title="E77">A466</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_19"><i><font color="Green" title="E78">A467</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:33_1_1_19"/>
<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="c15">g</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_19"><i><font color="Green" title="E78">A467</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_19"><i><font color="Green" title="E79">A468</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E7:33_1_1_19"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_19"><i><font color="Green" title="E77">A466</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_19"><i><font color="Green" title="E78">A467</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_19"><i><font color="Green" title="E79">A468</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_19"><i><font color="Green" title="E80">A469</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_19"><i><font color="Green" title="E81">A470</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E63:33_1_1"/><i><font color="Green" title="E77">A471</font></i>: 
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#K1" title="NAT_1:func.1">+</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="c13">f</font> holds <br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c13">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="c13">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="33_1_1_20"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">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> ;<br/>

<b>assume </b><a NAME="E1:33_1_1_20"/><i><font color="Green" title="E78">A472</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c13">f</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c17">i1</font>, <font color="Maroon" title="c18">j1</font>, <font color="Maroon" title="c19">i2</font>, <font color="Maroon" title="c20">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> <b> such that </b><br/><a NAME="E3:33_1_1_20"/><i><font color="Green" title="E79">A473</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">i1</font>,<font color="Maroon" title="c18">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>
 <b>and </b><br/><a NAME="E4:33_1_1_20"/><i><font color="Green" title="E80">A474</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">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="c17">i1</font>,<font color="Maroon" title="c18">j1</font>
 <b>and </b><br/><a NAME="E5:33_1_1_20"/><i><font color="Green" title="E81">A475</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i2</font>,<font color="Maroon" title="c20">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>
 <b>and </b><br/><a NAME="E6:33_1_1_20"/><i><font color="Green" title="E82">A476</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2</font>,<font color="Maroon" title="c20">j2</font>
 <b>and </b><br/><a NAME="E7:33_1_1_20"/><i><font color="Green" title="E83">A477</font></i>: 
( ( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> &amp; <font color="Maroon" title="c18">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j2</font> ) or ( <font color="Maroon" title="c17">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> &amp; <font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j2</font> ) or ( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j2</font> ) or ( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> &amp; <font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_20"><i><font color="Green" title="E78">A472</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/>
<a NAME="E8:33_1_1_20"/><i><font color="Green" title="E84">A478</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i1</font> &amp; <font color="Maroon" title="c17">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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">j1</font> &amp; <font color="Maroon" title="c18">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> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_20"><i><font color="Green" title="E79">A473</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/>
<a NAME="E9:33_1_1_20"/><i><font color="Green" title="E85">A479</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i2</font> &amp; <font color="Maroon" title="c19">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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">j2</font> &amp; <font color="Maroon" title="c20">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> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1_20"><i><font color="Green" title="E81">A475</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/>
<a NAME="E10:33_1_1_20"/><i><font color="Green" title="E86">A480</font></i>: 
( <font color="Maroon" title="c17">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c17">i1</font> &amp; <font color="Maroon" title="c18">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c18">j1</font> &amp; <font color="Maroon" title="c19">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c19">i2</font> &amp; <font color="Maroon" title="c20">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">j2</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_20_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_20_1"/>
( ( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> &amp; <font color="Maroon" title="c18">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j2</font> ) or ( <font color="Maroon" title="c17">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> &amp; <font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j2</font> ) or ( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j2</font> ) or ( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> &amp; <font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_20"><i><font color="Green" title="E83">A477</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_20_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_20_1_1"/>
( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> &amp; <font color="Maroon" title="c18">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_20_1_1"/><b>then </b>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c13">f</font>,<font color="Maroon" title="c16">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="c17">i1</font>,<font color="Maroon" title="c18">j1</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_20"><i><font color="Green" title="E78">A472</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_20"><i><font color="Green" title="E79">A473</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_20"><i><font color="Green" title="E80">A474</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_20"><i><font color="Green" title="E81">A475</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_20"><i><font color="Green" title="E82">A476</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_20"><i><font color="Green" title="E86">A480</font></i></a>, <a class="ref" href="gobrd13.html#D2" title="GOBRD13:def.2">GOBRD13:def 2</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_20_1_1"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c13">f</font>,<font color="Maroon" title="c16">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="c13">f</font>,<font color="Maroon" title="c16">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>
 <b>by </b><i><a class="txt" href="jordan9.html#E8:33_1_1_20"><i><font color="Green" title="E84">A478</font></i></a>, <a class="ref" href="gobrd11.html#T35" title="GOBRD11:th.35">GOBRD11:35</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_20_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_20_1_2"/><i><font color="Green" title="E87">A481</font></i>: 
( <font color="Maroon" title="c17">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> &amp; <font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_20_1_2"/><i><font color="Green" title="E88">A482</font></i>: 
<font color="Maroon" title="c18">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E8:33_1_1_20"><i><font color="Green" title="E84">A478</font></i></a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E3:33_1_1_20_1_2"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c13">f</font>,<font color="Maroon" title="c16">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="c17">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_20"><i><font color="Green" title="E78">A472</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_20"><i><font color="Green" title="E79">A473</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_20"><i><font color="Green" title="E80">A474</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_20"><i><font color="Green" title="E81">A475</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_20"><i><font color="Green" title="E82">A476</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_20"><i><font color="Green" title="E86">A480</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_20_1_2"><i><font color="Green" title="E87">A481</font></i></a>, <a class="ref" href="gobrd13.html#D2" title="GOBRD13:def.2">GOBRD13:def 2</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_20_1_2"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c13">f</font>,<font color="Maroon" title="c16">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="c13">f</font>,<font color="Maroon" title="c16">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>
 <b>by </b><i><a class="txt" href="jordan9.html#E8:33_1_1_20"><i><font color="Green" title="E84">A478</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_20_1_2"><i><font color="Green" title="E88">A482</font></i></a>, <a class="ref" href="gobrd11.html#T35" title="GOBRD11:th.35">GOBRD11:35</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_20_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_20_1_3"/>
( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_20_1_3"/><b>then </b>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c13">f</font>,<font color="Maroon" title="c16">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">i2</font>,<font color="Maroon" title="c20">j2</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_20"><i><font color="Green" title="E78">A472</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_20"><i><font color="Green" title="E79">A473</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_20"><i><font color="Green" title="E80">A474</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_20"><i><font color="Green" title="E81">A475</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_20"><i><font color="Green" title="E82">A476</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_20"><i><font color="Green" title="E86">A480</font></i></a>, <a class="ref" href="gobrd13.html#D2" title="GOBRD13:def.2">GOBRD13:def 2</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_20_1_3"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c13">f</font>,<font color="Maroon" title="c16">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="c13">f</font>,<font color="Maroon" title="c16">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>
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_20"><i><font color="Green" title="E85">A479</font></i></a>, <a class="ref" href="gobrd11.html#T35" title="GOBRD11:th.35">GOBRD11:35</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_20_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_20_1_4"/><i><font color="Green" title="E87">A483</font></i>: 
( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> &amp; <font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_20_1_4"/><i><font color="Green" title="E88">A484</font></i>: 
<font color="Maroon" title="c17">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E8:33_1_1_20"><i><font color="Green" title="E84">A478</font></i></a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E3:33_1_1_20_1_4"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c13">f</font>,<font color="Maroon" title="c16">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="c17">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c20">j2</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_20"><i><font color="Green" title="E78">A472</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_20"><i><font color="Green" title="E79">A473</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_20"><i><font color="Green" title="E80">A474</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_20"><i><font color="Green" title="E81">A475</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_20"><i><font color="Green" title="E82">A476</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_20"><i><font color="Green" title="E86">A480</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_20_1_4"><i><font color="Green" title="E87">A483</font></i></a>, <a class="ref" href="gobrd13.html#D2" title="GOBRD13:def.2">GOBRD13:def 2</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_20_1_4"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c13">f</font>,<font color="Maroon" title="c16">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="c13">f</font>,<font color="Maroon" title="c16">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>
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_20"><i><font color="Green" title="E85">A479</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_20_1_4"><i><font color="Green" title="E88">A484</font></i></a>, <a class="ref" href="gobrd11.html#T35" title="GOBRD11:th.35">GOBRD11:35</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E64:33_1_1"/><i><font color="Green" title="E78">A485</font></i>: 
<font color="Maroon" title="c15">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="33_1_1_21"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard5.html#D4" title="GOBOARD5:def.4">GOBOARD5:def 4</a><br/><b>let </b><font color="Maroon" title="c17">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> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_21"/><i><font color="Green" title="E79">A486</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">j</font>
 <b>and </b><br/><a NAME="E2:33_1_1_21"/><i><font color="Green" title="E80">A487</font></i>: 
( ( <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &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="c15">g</font> ) or <font color="Maroon" title="c17">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c15">g</font> )
 ;<br/>

<a NAME="E3:33_1_1_21"/><i><font color="Green" title="E81">A488</font></i>: 
<font color="Maroon" title="c17">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E4:33_1_1_21"/><b>then </b><i><font color="Green" title="E82">A489</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:33_1_1_21"/><i><font color="Green" title="E83">A490</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E6:33_1_1_21"/><i><font color="Green" title="E84">A491</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">j</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E7:33_1_1_21"/>
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">j</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E8:33_1_1_21"/><b>then </b><i><font color="Green" title="E85">A492</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21"><i><font color="Green" title="E81">A488</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_21_1"/>
( ( <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &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="c15">g</font> ) or ( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c17">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c15">g</font> ) or ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c17">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c15">g</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_21"><i><font color="Green" title="E80">A487</font></i></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1"/><i><font color="Green" title="E86">A493</font></i>: 
( <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &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="c15">g</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1"/><b>then </b><i><font color="Green" title="E87">A494</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c15">g</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:33_1_1_21_1_1"/><b>then </b><i><font color="Green" title="E88">A495</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_21_1_1"><i><font color="Green" title="E86">A493</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E4:33_1_1_21_1_1"/><i><font color="Green" title="E89">A496</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_21_1_1"><i><font color="Green" title="E86">A493</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:33_1_1_21_1_1"/><i><font color="Green" title="E90">A497</font></i>: 
<font color="Maroon" title="c16">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="c15">g</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_21_1_1"><i><font color="Green" title="E87">A494</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>consider </b><font color="Maroon" title="c18">i1</font>, <font color="Maroon" title="c19">j1</font>, <font color="Maroon" title="c20">i2</font>, <font color="Maroon" title="c21">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> <b> such that </b><br/><a NAME="E7:33_1_1_21_1_1"/><i><font color="Green" title="E91">A498</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i1</font>,<font color="Maroon" title="c19">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>
 <b>and </b><br/><a NAME="E8:33_1_1_21_1_1"/><i><font color="Green" title="E92">A499</font></i>: 
<font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">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="c18">i1</font>,<font color="Maroon" title="c19">j1</font>
 <b>and </b><br/><a NAME="E9:33_1_1_21_1_1"/><i><font color="Green" title="E93">A500</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i2</font>,<font color="Maroon" title="c21">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>
 <b>and </b><br/><a NAME="E10:33_1_1_21_1_1"/><i><font color="Green" title="E94">A501</font></i>: 
<font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2</font>,<font color="Maroon" title="c21">j2</font>
 <b>and </b><br/><a NAME="E11:33_1_1_21_1_1"/><i><font color="Green" title="E95">A502</font></i>: 
( ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E58:33_1_1"><i><font color="Green" title="E72">A432</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1"><i><font color="Green" title="E86">A493</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_21_1_1"><i><font color="Green" title="E87">A494</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E12:33_1_1_21_1_1"/><i><font color="Green" title="E96">A503</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i1</font> &amp; <font color="Maroon" title="c18">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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">j1</font> &amp; <font color="Maroon" title="c19">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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_21_1_1"><i><font color="Green" title="E91">A498</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E13:33_1_1_21_1_1"/><i><font color="Green" title="E97">A504</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c20">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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c21">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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_21_1_1"><i><font color="Green" title="E93">A500</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E14:33_1_1_21_1_1"/><i><font color="Green" title="E98">A505</font></i>: 
<font color="Maroon" title="c17">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c15">g</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_21_1_1"><i><font color="Green" title="E86">A493</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E15:33_1_1_21_1_1"/><b>then </b><i><font color="Green" title="E99">A506</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c17">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c17">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c22">i1'</font>, <font color="Maroon" title="c23">j1'</font>, <font color="Maroon" title="c24">i2'</font>, <font color="Maroon" title="c25">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> <b> such that </b><br/><a NAME="E17:33_1_1_21_1_1"/><i><font color="Green" title="E100">A507</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">i1'</font>,<font color="Maroon" title="c23">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>
 <b>and </b><br/><a NAME="E18:33_1_1_21_1_1"/><i><font color="Green" title="E101">A508</font></i>: 
<font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c17">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="c22">i1'</font>,<font color="Maroon" title="c23">j1'</font>
 <b>and </b><br/><a NAME="E19:33_1_1_21_1_1"/><i><font color="Green" title="E102">A509</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">i2'</font>,<font color="Maroon" title="c25">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>
 <b>and </b><br/><a NAME="E20:33_1_1_21_1_1"/><i><font color="Green" title="E103">A510</font></i>: 
<font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c24">i2'</font>,<font color="Maroon" title="c25">j2'</font>
 <b>and </b><br/><a NAME="E21:33_1_1_21_1_1"/><i><font color="Green" title="E104">A511</font></i>: 
( ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E58:33_1_1"><i><font color="Green" title="E72">A432</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_1"><i><font color="Green" title="E98">A505</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E22:33_1_1_21_1_1"/><i><font color="Green" title="E105">A512</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">i1'</font> &amp; <font color="Maroon" title="c22">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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">j1'</font> &amp; <font color="Maroon" title="c23">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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E17:33_1_1_21_1_1"><i><font color="Green" title="E100">A507</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E23:33_1_1_21_1_1"/><i><font color="Green" title="E106">A513</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c24">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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">j2'</font> &amp; <font color="Maroon" title="c25">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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1_21_1_1"><i><font color="Green" title="E102">A509</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><b>assume </b><a NAME="E24:33_1_1_21_1_1"/><i><font color="Green" title="E107">A514</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c16">i</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c17">j</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_21_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_21_1_1_1_1"/>
( ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) or ( <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_21_1_1"><i><font color="Green" title="E95">A502</font></i></a>, <a class="txt" href="jordan9.html#E21:33_1_1_21_1_1"><i><font color="Green" title="E104">A511</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_1"/><i><font color="Green" title="E108">A515</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E109">A516</font></i>: 
<font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_1"><i><font color="Green" title="E88">A495</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_21_1_1"><i><font color="Green" title="E96">A503</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_1"><i><font color="Green" title="E97">A504</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_1"><i><font color="Green" title="E99">A506</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_1"><i><font color="Green" title="E105">A512</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_1"><i><font color="Green" title="E106">A513</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_1"><i><font color="Green" title="E107">A514</font></i></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></i>;<br/><a NAME="E3:33_1_1_21_1_1_1_1_1"/>
( <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> or <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_1"><i><font color="Green" title="E88">A495</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_21_1_1"><i><font color="Green" title="E96">A503</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_1"><i><font color="Green" title="E97">A504</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_1"><i><font color="Green" title="E99">A506</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_1"><i><font color="Green" title="E105">A512</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_1"><i><font color="Green" title="E106">A513</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_1"><i><font color="Green" title="E107">A514</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1_1_1_1"><i><font color="Green" title="E108">A515</font></i></a>, <a class="ref" href="goboard7.html#T24" title="GOBOARD7:th.24">GOBOARD7:24</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E61:33_1_1"><i><font color="Green" title="E75">A457</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_21"><i><font color="Green" title="E81">A488</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21"><i><font color="Green" title="E83">A490</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21"><i><font color="Green" title="E85">A492</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1"><i><font color="Green" title="E86">A493</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_21_1_1"><i><font color="Green" title="E87">A494</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21_1_1"><i><font color="Green" title="E90">A497</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_1"><i><font color="Green" title="E98">A505</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1_1_1_1"><i><font color="Green" title="E108">A515</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_21_1_1_1_1_1"><i><font color="Green" title="E109">A516</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_2"/><i><font color="Green" title="E108">A517</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_2"/><b>then </b>
( ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_1"><i><font color="Green" title="E88">A495</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_21_1_1"><i><font color="Green" title="E96">A503</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_1"><i><font color="Green" title="E97">A504</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_1"><i><font color="Green" title="E99">A506</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_1"><i><font color="Green" title="E105">A512</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_1"><i><font color="Green" title="E106">A513</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_1"><i><font color="Green" title="E107">A514</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E61:33_1_1"><i><font color="Green" title="E75">A457</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21"><i><font color="Green" title="E85">A492</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1"><i><font color="Green" title="E86">A493</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_1"><i><font color="Green" title="E89">A496</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21_1_1"><i><font color="Green" title="E90">A497</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_1"><i><font color="Green" title="E98">A505</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1_1_1_2"><i><font color="Green" title="E108">A517</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_3"/><i><font color="Green" title="E108">A518</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_3"/><b>then </b>
( ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_1"><i><font color="Green" title="E88">A495</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_21_1_1"><i><font color="Green" title="E96">A503</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_1"><i><font color="Green" title="E97">A504</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_1"><i><font color="Green" title="E99">A506</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_1"><i><font color="Green" title="E105">A512</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_1"><i><font color="Green" title="E106">A513</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_1"><i><font color="Green" title="E107">A514</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E61:33_1_1"><i><font color="Green" title="E75">A457</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21"><i><font color="Green" title="E85">A492</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1"><i><font color="Green" title="E86">A493</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_1"><i><font color="Green" title="E89">A496</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21_1_1"><i><font color="Green" title="E90">A497</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_1"><i><font color="Green" title="E98">A505</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1_1_1_3"><i><font color="Green" title="E108">A518</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_4"/><i><font color="Green" title="E108">A519</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_4"/><b>then </b><i><font color="Green" title="E109">A520</font></i>: 
<font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_1"><i><font color="Green" title="E88">A495</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_21_1_1"><i><font color="Green" title="E96">A503</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_1"><i><font color="Green" title="E97">A504</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_1"><i><font color="Green" title="E99">A506</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_1"><i><font color="Green" title="E105">A512</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_1"><i><font color="Green" title="E106">A513</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_1"><i><font color="Green" title="E107">A514</font></i></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></i>;<br/><a NAME="E3:33_1_1_21_1_1_1_1_4"/>
( <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> or <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_1"><i><font color="Green" title="E88">A495</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_21_1_1"><i><font color="Green" title="E96">A503</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_1"><i><font color="Green" title="E97">A504</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_1"><i><font color="Green" title="E99">A506</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_1"><i><font color="Green" title="E105">A512</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_1"><i><font color="Green" title="E106">A513</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_1"><i><font color="Green" title="E107">A514</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1_1_1_4"><i><font color="Green" title="E108">A519</font></i></a>, <a class="ref" href="goboard7.html#T24" title="GOBOARD7:th.24">GOBOARD7:24</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_1_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E61:33_1_1"><i><font color="Green" title="E75">A457</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1"><i><font color="Green" title="E86">A493</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_1"><i><font color="Green" title="E89">A496</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21_1_1"><i><font color="Green" title="E90">A497</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_1"><i><font color="Green" title="E98">A505</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1_1_1_4"><i><font color="Green" title="E108">A519</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_21_1_1_1_1_4"><i><font color="Green" title="E109">A520</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_5"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_5"/><i><font color="Green" title="E108">A521</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_5"/><b>then </b>
( ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_1"><i><font color="Green" title="E88">A495</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_21_1_1"><i><font color="Green" title="E96">A503</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_1"><i><font color="Green" title="E97">A504</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_1"><i><font color="Green" title="E99">A506</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_1"><i><font color="Green" title="E105">A512</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_1"><i><font color="Green" title="E106">A513</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_1"><i><font color="Green" title="E107">A514</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_1_1_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E61:33_1_1"><i><font color="Green" title="E75">A457</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21"><i><font color="Green" title="E85">A492</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1"><i><font color="Green" title="E86">A493</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_1"><i><font color="Green" title="E89">A496</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21_1_1"><i><font color="Green" title="E90">A497</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_1"><i><font color="Green" title="E98">A505</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1_1_1_5"><i><font color="Green" title="E108">A521</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_6"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_6"/><i><font color="Green" title="E108">A522</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_6"/><b>then </b><i><font color="Green" title="E109">A523</font></i>: 
<font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_1"><i><font color="Green" title="E88">A495</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_21_1_1"><i><font color="Green" title="E96">A503</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_1"><i><font color="Green" title="E97">A504</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_1"><i><font color="Green" title="E99">A506</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_1"><i><font color="Green" title="E105">A512</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_1"><i><font color="Green" title="E106">A513</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_1"><i><font color="Green" title="E107">A514</font></i></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></i>;<br/><a NAME="E3:33_1_1_21_1_1_1_1_6"/>
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> or <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_1"><i><font color="Green" title="E88">A495</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_21_1_1"><i><font color="Green" title="E96">A503</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_1"><i><font color="Green" title="E97">A504</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_1"><i><font color="Green" title="E99">A506</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_1"><i><font color="Green" title="E105">A512</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_1"><i><font color="Green" title="E106">A513</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_1"><i><font color="Green" title="E107">A514</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1_1_1_6"><i><font color="Green" title="E108">A522</font></i></a>, <a class="ref" href="goboard7.html#T25" title="GOBOARD7:th.25">GOBOARD7:25</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_1_1_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E61:33_1_1"><i><font color="Green" title="E75">A457</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_21"><i><font color="Green" title="E81">A488</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21"><i><font color="Green" title="E83">A490</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21"><i><font color="Green" title="E85">A492</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1"><i><font color="Green" title="E86">A493</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_21_1_1"><i><font color="Green" title="E87">A494</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21_1_1"><i><font color="Green" title="E90">A497</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_1"><i><font color="Green" title="E98">A505</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1_1_1_6"><i><font color="Green" title="E108">A522</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_21_1_1_1_1_6"><i><font color="Green" title="E109">A523</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_7"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_7"/><i><font color="Green" title="E108">A524</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_7"/><b>then </b><i><font color="Green" title="E109">A525</font></i>: 
<font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_1"><i><font color="Green" title="E88">A495</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_21_1_1"><i><font color="Green" title="E96">A503</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_1"><i><font color="Green" title="E97">A504</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_1"><i><font color="Green" title="E99">A506</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_1"><i><font color="Green" title="E105">A512</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_1"><i><font color="Green" title="E106">A513</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_1"><i><font color="Green" title="E107">A514</font></i></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></i>;<br/><a NAME="E3:33_1_1_21_1_1_1_1_7"/>
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> or <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_1"><i><font color="Green" title="E88">A495</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_21_1_1"><i><font color="Green" title="E96">A503</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_1"><i><font color="Green" title="E97">A504</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_1"><i><font color="Green" title="E99">A506</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_1"><i><font color="Green" title="E105">A512</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_1"><i><font color="Green" title="E106">A513</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_1"><i><font color="Green" title="E107">A514</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1_1_1_7"><i><font color="Green" title="E108">A524</font></i></a>, <a class="ref" href="goboard7.html#T25" title="GOBOARD7:th.25">GOBOARD7:25</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_1_1_1_7"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E61:33_1_1"><i><font color="Green" title="E75">A457</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21"><i><font color="Green" title="E85">A492</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1"><i><font color="Green" title="E86">A493</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_1"><i><font color="Green" title="E89">A496</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21_1_1"><i><font color="Green" title="E90">A497</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_1"><i><font color="Green" title="E98">A505</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1_1_1_7"><i><font color="Green" title="E108">A524</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_21_1_1_1_1_7"><i><font color="Green" title="E109">A525</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_8"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_8"/><i><font color="Green" title="E108">A526</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_8"/><b>then </b>
( ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1</font> &amp; <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_1"><i><font color="Green" title="E88">A495</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_21_1_1"><i><font color="Green" title="E96">A503</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_1"><i><font color="Green" title="E97">A504</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_1"><i><font color="Green" title="E99">A506</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_1"><i><font color="Green" title="E105">A512</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_1"><i><font color="Green" title="E106">A513</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_1"><i><font color="Green" title="E107">A514</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_1_1_1_8"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E61:33_1_1"><i><font color="Green" title="E75">A457</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21"><i><font color="Green" title="E85">A492</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1"><i><font color="Green" title="E86">A493</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_1"><i><font color="Green" title="E89">A496</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21_1_1"><i><font color="Green" title="E90">A497</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_1"><i><font color="Green" title="E98">A505</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1_1_1_8"><i><font color="Green" title="E108">A526</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_9"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_9"/><i><font color="Green" title="E108">A527</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_9"/><b>then </b>
( ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_1"><i><font color="Green" title="E88">A495</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_21_1_1"><i><font color="Green" title="E96">A503</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_1"><i><font color="Green" title="E97">A504</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_1"><i><font color="Green" title="E99">A506</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_1"><i><font color="Green" title="E105">A512</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_1"><i><font color="Green" title="E106">A513</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_1"><i><font color="Green" title="E107">A514</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_1_1_1_9"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E61:33_1_1"><i><font color="Green" title="E75">A457</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21"><i><font color="Green" title="E85">A492</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1"><i><font color="Green" title="E86">A493</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_1"><i><font color="Green" title="E89">A496</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21_1_1"><i><font color="Green" title="E90">A497</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_1"><i><font color="Green" title="E98">A505</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1_1_1_9"><i><font color="Green" title="E108">A527</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_10"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_10"/><i><font color="Green" title="E108">A528</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_10"/><b>then </b><i><font color="Green" title="E109">A529</font></i>: 
<font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_1"><i><font color="Green" title="E88">A495</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_21_1_1"><i><font color="Green" title="E96">A503</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_1"><i><font color="Green" title="E97">A504</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_1"><i><font color="Green" title="E99">A506</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_1"><i><font color="Green" title="E105">A512</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_1"><i><font color="Green" title="E106">A513</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_1"><i><font color="Green" title="E107">A514</font></i></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></i>;<br/><a NAME="E3:33_1_1_21_1_1_1_1_10"/>
( <font color="Maroon" title="c20">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> or <font color="Maroon" title="c20">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_1"><i><font color="Green" title="E88">A495</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_21_1_1"><i><font color="Green" title="E96">A503</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_1"><i><font color="Green" title="E97">A504</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_1"><i><font color="Green" title="E99">A506</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_1"><i><font color="Green" title="E105">A512</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_1"><i><font color="Green" title="E106">A513</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_1"><i><font color="Green" title="E107">A514</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1_1_1_10"><i><font color="Green" title="E108">A528</font></i></a>, <a class="ref" href="goboard7.html#T25" title="GOBOARD7:th.25">GOBOARD7:25</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_1_1_1_10"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E61:33_1_1"><i><font color="Green" title="E75">A457</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1"><i><font color="Green" title="E86">A493</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_1"><i><font color="Green" title="E89">A496</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21_1_1"><i><font color="Green" title="E90">A497</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_1"><i><font color="Green" title="E98">A505</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1_1_1_10"><i><font color="Green" title="E108">A528</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_21_1_1_1_1_10"><i><font color="Green" title="E109">A529</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_11"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_11"/><i><font color="Green" title="E108">A530</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_11"/><b>then </b><i><font color="Green" title="E109">A531</font></i>: 
<font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_1"><i><font color="Green" title="E88">A495</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_21_1_1"><i><font color="Green" title="E96">A503</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_1"><i><font color="Green" title="E97">A504</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_1"><i><font color="Green" title="E99">A506</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_1"><i><font color="Green" title="E105">A512</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_1"><i><font color="Green" title="E106">A513</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_1"><i><font color="Green" title="E107">A514</font></i></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></i>;<br/><a NAME="E3:33_1_1_21_1_1_1_1_11"/>
( <font color="Maroon" title="c20">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> or <font color="Maroon" title="c20">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_1"><i><font color="Green" title="E88">A495</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_21_1_1"><i><font color="Green" title="E96">A503</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_1"><i><font color="Green" title="E97">A504</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_1"><i><font color="Green" title="E99">A506</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_1"><i><font color="Green" title="E105">A512</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_1"><i><font color="Green" title="E106">A513</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_1"><i><font color="Green" title="E107">A514</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1_1_1_11"><i><font color="Green" title="E108">A530</font></i></a>, <a class="ref" href="goboard7.html#T25" title="GOBOARD7:th.25">GOBOARD7:25</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_1_1_1_11"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E61:33_1_1"><i><font color="Green" title="E75">A457</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21"><i><font color="Green" title="E85">A492</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1"><i><font color="Green" title="E86">A493</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_1"><i><font color="Green" title="E89">A496</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_1"><i><font color="Green" title="E98">A505</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1_1_1_11"><i><font color="Green" title="E108">A530</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_21_1_1_1_1_11"><i><font color="Green" title="E109">A531</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_12"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_12"/><i><font color="Green" title="E108">A532</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_12"/><b>then </b>
( ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c25">j2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c25">j2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_1"><i><font color="Green" title="E88">A495</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_21_1_1"><i><font color="Green" title="E96">A503</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_1"><i><font color="Green" title="E97">A504</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_1"><i><font color="Green" title="E99">A506</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_1"><i><font color="Green" title="E105">A512</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_1"><i><font color="Green" title="E106">A513</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_1"><i><font color="Green" title="E107">A514</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_1_1_1_12"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E61:33_1_1"><i><font color="Green" title="E75">A457</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21"><i><font color="Green" title="E85">A492</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1"><i><font color="Green" title="E86">A493</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_1"><i><font color="Green" title="E89">A496</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21_1_1"><i><font color="Green" title="E90">A497</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_1"><i><font color="Green" title="E98">A505</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1_1_1_12"><i><font color="Green" title="E108">A532</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_13"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_13"/><i><font color="Green" title="E108">A533</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_13"/><b>then </b><i><font color="Green" title="E109">A534</font></i>: 
<font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_1"><i><font color="Green" title="E88">A495</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_21_1_1"><i><font color="Green" title="E96">A503</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_1"><i><font color="Green" title="E97">A504</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_1"><i><font color="Green" title="E99">A506</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_1"><i><font color="Green" title="E105">A512</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_1"><i><font color="Green" title="E106">A513</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_1"><i><font color="Green" title="E107">A514</font></i></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></i>;<br/><a NAME="E3:33_1_1_21_1_1_1_1_13"/>
( <font color="Maroon" title="c21">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> or <font color="Maroon" title="c21">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_1"><i><font color="Green" title="E88">A495</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_21_1_1"><i><font color="Green" title="E96">A503</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_1"><i><font color="Green" title="E97">A504</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_1"><i><font color="Green" title="E99">A506</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_1"><i><font color="Green" title="E105">A512</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_1"><i><font color="Green" title="E106">A513</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_1"><i><font color="Green" title="E107">A514</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1_1_1_13"><i><font color="Green" title="E108">A533</font></i></a>, <a class="ref" href="goboard7.html#T24" title="GOBOARD7:th.24">GOBOARD7:24</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_1_1_1_13"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E61:33_1_1"><i><font color="Green" title="E75">A457</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1"><i><font color="Green" title="E86">A493</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_1"><i><font color="Green" title="E89">A496</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21_1_1"><i><font color="Green" title="E90">A497</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_1"><i><font color="Green" title="E98">A505</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1_1_1_13"><i><font color="Green" title="E108">A533</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_21_1_1_1_1_13"><i><font color="Green" title="E109">A534</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_14"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_14"/><i><font color="Green" title="E108">A535</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_14"/><b>then </b>
( ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> &amp; <font color="Maroon" title="c21">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> &amp; <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_1"><i><font color="Green" title="E88">A495</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_21_1_1"><i><font color="Green" title="E96">A503</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_1"><i><font color="Green" title="E97">A504</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_1"><i><font color="Green" title="E99">A506</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_1"><i><font color="Green" title="E105">A512</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_1"><i><font color="Green" title="E106">A513</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_1"><i><font color="Green" title="E107">A514</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_1_1_1_14"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E61:33_1_1"><i><font color="Green" title="E75">A457</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21"><i><font color="Green" title="E85">A492</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1"><i><font color="Green" title="E86">A493</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_1"><i><font color="Green" title="E89">A496</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21_1_1"><i><font color="Green" title="E90">A497</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_1"><i><font color="Green" title="E98">A505</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1_1_1_14"><i><font color="Green" title="E108">A535</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_15"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_15"/><i><font color="Green" title="E108">A536</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_15"/><b>then </b>
( ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c21">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_1"><i><font color="Green" title="E88">A495</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_21_1_1"><i><font color="Green" title="E96">A503</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_1"><i><font color="Green" title="E97">A504</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_1"><i><font color="Green" title="E99">A506</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_1"><i><font color="Green" title="E105">A512</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_1"><i><font color="Green" title="E106">A513</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_1"><i><font color="Green" title="E107">A514</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_1_1_1_15"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E61:33_1_1"><i><font color="Green" title="E75">A457</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21"><i><font color="Green" title="E85">A492</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1"><i><font color="Green" title="E86">A493</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_1"><i><font color="Green" title="E89">A496</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21_1_1"><i><font color="Green" title="E90">A497</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_1"><i><font color="Green" title="E98">A505</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1_1_1_15"><i><font color="Green" title="E108">A536</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_16"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_16"/><i><font color="Green" title="E108">A537</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_16"/><b>then </b><i><font color="Green" title="E109">A538</font></i>: 
<font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_1"><i><font color="Green" title="E88">A495</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_21_1_1"><i><font color="Green" title="E96">A503</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_1"><i><font color="Green" title="E97">A504</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_1"><i><font color="Green" title="E99">A506</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_1"><i><font color="Green" title="E105">A512</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_1"><i><font color="Green" title="E106">A513</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_1"><i><font color="Green" title="E107">A514</font></i></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></i>;<br/><a NAME="E3:33_1_1_21_1_1_1_1_16"/>
( <font color="Maroon" title="c21">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> or <font color="Maroon" title="c21">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_1"><i><font color="Green" title="E88">A495</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_21_1_1"><i><font color="Green" title="E96">A503</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_1"><i><font color="Green" title="E97">A504</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_1"><i><font color="Green" title="E99">A506</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_1"><i><font color="Green" title="E105">A512</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_1"><i><font color="Green" title="E106">A513</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_1"><i><font color="Green" title="E107">A514</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1_1_1_16"><i><font color="Green" title="E108">A537</font></i></a>, <a class="ref" href="goboard7.html#T24" title="GOBOARD7:th.24">GOBOARD7:24</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_1_1_1_16"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E61:33_1_1"><i><font color="Green" title="E75">A457</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21"><i><font color="Green" title="E85">A492</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1"><i><font color="Green" title="E86">A493</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_1"><i><font color="Green" title="E89">A496</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21_1_1"><i><font color="Green" title="E92">A499</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_21_1_1"><i><font color="Green" title="E94">A501</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_1"><i><font color="Green" title="E98">A505</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_1"><i><font color="Green" title="E101">A508</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_1"><i><font color="Green" title="E103">A510</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_1_1_1_16"><i><font color="Green" title="E108">A537</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_21_1_1_1_1_16"><i><font color="Green" title="E109">A538</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E26:33_1_1_21_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_2"/>
( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c17">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c15">g</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_2"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c16">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c17">j</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T65" title="XBOOLE_1:th.65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3"/><i><font color="Green" title="E86">A539</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c17">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c15">g</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3"/><b>then </b><i><font color="Green" title="E87">A540</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c15">g</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:33_1_1_21_1_3"/><b>then </b><i><font color="Green" title="E88">A541</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_21_1_3"><i><font color="Green" title="E86">A539</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E4:33_1_1_21_1_3"/><i><font color="Green" title="E89">A542</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_21_1_3"><i><font color="Green" title="E86">A539</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:33_1_1_21_1_3"/><i><font color="Green" title="E90">A543</font></i>: 
<font color="Maroon" title="c16">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="c15">g</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_21_1_3"><i><font color="Green" title="E87">A540</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E6:33_1_1_21_1_3"/><i><font color="Green" title="E91">A544</font></i>: 
<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="c15">g</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_21_1_3"><i><font color="Green" title="E86">A539</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><b>consider </b><font color="Maroon" title="c18">i1</font>, <font color="Maroon" title="c19">j1</font>, <font color="Maroon" title="c20">i2</font>, <font color="Maroon" title="c21">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> <b> such that </b><br/><a NAME="E8:33_1_1_21_1_3"/><i><font color="Green" title="E92">A545</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i1</font>,<font color="Maroon" title="c19">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>
 <b>and </b><br/><a NAME="E9:33_1_1_21_1_3"/><i><font color="Green" title="E93">A546</font></i>: 
<font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">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="c18">i1</font>,<font color="Maroon" title="c19">j1</font>
 <b>and </b><br/><a NAME="E10:33_1_1_21_1_3"/><i><font color="Green" title="E94">A547</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i2</font>,<font color="Maroon" title="c21">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>
 <b>and </b><br/><a NAME="E11:33_1_1_21_1_3"/><i><font color="Green" title="E95">A548</font></i>: 
<font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">i2</font>,<font color="Maroon" title="c21">j2</font>
 <b>and </b><br/><a NAME="E12:33_1_1_21_1_3"/><i><font color="Green" title="E96">A549</font></i>: 
( ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E58:33_1_1"><i><font color="Green" title="E72">A432</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3"><i><font color="Green" title="E86">A539</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_21_1_3"><i><font color="Green" title="E87">A540</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E13:33_1_1_21_1_3"/><i><font color="Green" title="E97">A550</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i1</font> &amp; <font color="Maroon" title="c18">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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">j1</font> &amp; <font color="Maroon" title="c19">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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E8:33_1_1_21_1_3"><i><font color="Green" title="E92">A545</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E14:33_1_1_21_1_3"/><i><font color="Green" title="E98">A551</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c20">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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c21">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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E10:33_1_1_21_1_3"><i><font color="Green" title="E94">A547</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E15:33_1_1_21_1_3"/><i><font color="Green" title="E99">A552</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c17">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c17">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3"><i><font color="Green" title="E86">A539</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c22">i1'</font>, <font color="Maroon" title="c23">j1'</font>, <font color="Maroon" title="c24">i2'</font>, <font color="Maroon" title="c25">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> <b> such that </b><br/><a NAME="E17:33_1_1_21_1_3"/><i><font color="Green" title="E100">A553</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">i1'</font>,<font color="Maroon" title="c23">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>
 <b>and </b><br/><a NAME="E18:33_1_1_21_1_3"/><i><font color="Green" title="E101">A554</font></i>: 
<font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c17">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="c22">i1'</font>,<font color="Maroon" title="c23">j1'</font>
 <b>and </b><br/><a NAME="E19:33_1_1_21_1_3"/><i><font color="Green" title="E102">A555</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">i2'</font>,<font color="Maroon" title="c25">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>
 <b>and </b><br/><a NAME="E20:33_1_1_21_1_3"/><i><font color="Green" title="E103">A556</font></i>: 
<font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c24">i2'</font>,<font color="Maroon" title="c25">j2'</font>
 <b>and </b><br/><a NAME="E21:33_1_1_21_1_3"/><i><font color="Green" title="E104">A557</font></i>: 
( ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E58:33_1_1"><i><font color="Green" title="E72">A432</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3"><i><font color="Green" title="E86">A539</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E22:33_1_1_21_1_3"/><i><font color="Green" title="E105">A558</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">i1'</font> &amp; <font color="Maroon" title="c22">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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">j1'</font> &amp; <font color="Maroon" title="c23">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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E17:33_1_1_21_1_3"><i><font color="Green" title="E100">A553</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E23:33_1_1_21_1_3"/><i><font color="Green" title="E106">A559</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c24">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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">j2'</font> &amp; <font color="Maroon" title="c25">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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1_21_1_3"><i><font color="Green" title="E102">A555</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><b>assume </b><a NAME="E24:33_1_1_21_1_3"/><i><font color="Green" title="E107">A560</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c16">i</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c17">j</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_21_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_21_1_3_1_1"/>
( ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) or ( <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E12:33_1_1_21_1_3"><i><font color="Green" title="E96">A549</font></i></a>, <a class="txt" href="jordan9.html#E21:33_1_1_21_1_3"><i><font color="Green" title="E104">A557</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_1"/><i><font color="Green" title="E108">A561</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E109">A562</font></i>: 
<font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_3"><i><font color="Green" title="E88">A541</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_3"><i><font color="Green" title="E97">A550</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_3"><i><font color="Green" title="E98">A551</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_3"><i><font color="Green" title="E99">A552</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_3"><i><font color="Green" title="E105">A558</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_3"><i><font color="Green" title="E106">A559</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_3"><i><font color="Green" title="E107">A560</font></i></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></i>;<br/><a NAME="E3:33_1_1_21_1_3_1_1_1"/>
( <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> or <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_3"><i><font color="Green" title="E88">A541</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_3"><i><font color="Green" title="E97">A550</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_3"><i><font color="Green" title="E98">A551</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_3"><i><font color="Green" title="E99">A552</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_3"><i><font color="Green" title="E105">A558</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_3"><i><font color="Green" title="E106">A559</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_3"><i><font color="Green" title="E107">A560</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3_1_1_1"><i><font color="Green" title="E108">A561</font></i></a>, <a class="ref" href="goboard7.html#T24" title="GOBOARD7:th.24">GOBOARD7:24</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E62:33_1_1"><i><font color="Green" title="E76">A465</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21"><i><font color="Green" title="E83">A490</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21"><i><font color="Green" title="E85">A492</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3"><i><font color="Green" title="E86">A539</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21_1_3"><i><font color="Green" title="E91">A544</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3_1_1_1"><i><font color="Green" title="E108">A561</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_21_1_3_1_1_1"><i><font color="Green" title="E109">A562</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_2"/><i><font color="Green" title="E108">A563</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_2"/><b>then </b>
( ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_3"><i><font color="Green" title="E88">A541</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_3"><i><font color="Green" title="E97">A550</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_3"><i><font color="Green" title="E98">A551</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_3"><i><font color="Green" title="E99">A552</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_3"><i><font color="Green" title="E105">A558</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_3"><i><font color="Green" title="E106">A559</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_3"><i><font color="Green" title="E107">A560</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E62:33_1_1"><i><font color="Green" title="E76">A465</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21"><i><font color="Green" title="E85">A492</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3"><i><font color="Green" title="E86">A539</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_3"><i><font color="Green" title="E89">A542</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21_1_3"><i><font color="Green" title="E90">A543</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21_1_3"><i><font color="Green" title="E91">A544</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3_1_1_2"><i><font color="Green" title="E108">A563</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_3"/><i><font color="Green" title="E108">A564</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_3"/><b>then </b>
( ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_3"><i><font color="Green" title="E88">A541</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_3"><i><font color="Green" title="E97">A550</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_3"><i><font color="Green" title="E98">A551</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_3"><i><font color="Green" title="E99">A552</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_3"><i><font color="Green" title="E105">A558</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_3"><i><font color="Green" title="E106">A559</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_3"><i><font color="Green" title="E107">A560</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_3_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E62:33_1_1"><i><font color="Green" title="E76">A465</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21"><i><font color="Green" title="E85">A492</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3"><i><font color="Green" title="E86">A539</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_3"><i><font color="Green" title="E89">A542</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21_1_3"><i><font color="Green" title="E90">A543</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21_1_3"><i><font color="Green" title="E91">A544</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3_1_1_3"><i><font color="Green" title="E108">A564</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_4"/><i><font color="Green" title="E108">A565</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_4"/><b>then </b><i><font color="Green" title="E109">A566</font></i>: 
<font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_3"><i><font color="Green" title="E88">A541</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_3"><i><font color="Green" title="E97">A550</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_3"><i><font color="Green" title="E98">A551</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_3"><i><font color="Green" title="E99">A552</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_3"><i><font color="Green" title="E105">A558</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_3"><i><font color="Green" title="E106">A559</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_3"><i><font color="Green" title="E107">A560</font></i></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></i>;<br/><a NAME="E3:33_1_1_21_1_3_1_1_4"/>
( <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> or <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c19">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_3"><i><font color="Green" title="E88">A541</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_3"><i><font color="Green" title="E97">A550</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_3"><i><font color="Green" title="E98">A551</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_3"><i><font color="Green" title="E99">A552</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_3"><i><font color="Green" title="E105">A558</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_3"><i><font color="Green" title="E106">A559</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_3"><i><font color="Green" title="E107">A560</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3_1_1_4"><i><font color="Green" title="E108">A565</font></i></a>, <a class="ref" href="goboard7.html#T24" title="GOBOARD7:th.24">GOBOARD7:24</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_3_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E61:33_1_1"><i><font color="Green" title="E75">A457</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3"><i><font color="Green" title="E86">A539</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_3"><i><font color="Green" title="E89">A542</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21_1_3"><i><font color="Green" title="E90">A543</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21_1_3"><i><font color="Green" title="E91">A544</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3_1_1_4"><i><font color="Green" title="E108">A565</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_21_1_3_1_1_4"><i><font color="Green" title="E109">A566</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_5"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_5"/><i><font color="Green" title="E108">A567</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_5"/><b>then </b>
( ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_3"><i><font color="Green" title="E88">A541</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_3"><i><font color="Green" title="E97">A550</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_3"><i><font color="Green" title="E98">A551</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_3"><i><font color="Green" title="E99">A552</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_3"><i><font color="Green" title="E105">A558</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_3"><i><font color="Green" title="E106">A559</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_3"><i><font color="Green" title="E107">A560</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_3_1_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E62:33_1_1"><i><font color="Green" title="E76">A465</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21"><i><font color="Green" title="E83">A490</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21"><i><font color="Green" title="E85">A492</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3"><i><font color="Green" title="E86">A539</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21_1_3"><i><font color="Green" title="E90">A543</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21_1_3"><i><font color="Green" title="E91">A544</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3_1_1_5"><i><font color="Green" title="E108">A567</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_6"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_6"/><i><font color="Green" title="E108">A568</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_6"/><b>then </b><i><font color="Green" title="E109">A569</font></i>: 
<font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_3"><i><font color="Green" title="E88">A541</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_3"><i><font color="Green" title="E97">A550</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_3"><i><font color="Green" title="E98">A551</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_3"><i><font color="Green" title="E99">A552</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_3"><i><font color="Green" title="E105">A558</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_3"><i><font color="Green" title="E106">A559</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_3"><i><font color="Green" title="E107">A560</font></i></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></i>;<br/><a NAME="E3:33_1_1_21_1_3_1_1_6"/>
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> or <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_3"><i><font color="Green" title="E88">A541</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_3"><i><font color="Green" title="E97">A550</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_3"><i><font color="Green" title="E98">A551</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_3"><i><font color="Green" title="E99">A552</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_3"><i><font color="Green" title="E105">A558</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_3"><i><font color="Green" title="E106">A559</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_3"><i><font color="Green" title="E107">A560</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3_1_1_6"><i><font color="Green" title="E108">A568</font></i></a>, <a class="ref" href="goboard7.html#T25" title="GOBOARD7:th.25">GOBOARD7:25</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_3_1_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E62:33_1_1"><i><font color="Green" title="E76">A465</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21"><i><font color="Green" title="E83">A490</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21"><i><font color="Green" title="E85">A492</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3"><i><font color="Green" title="E86">A539</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21_1_3"><i><font color="Green" title="E91">A544</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3_1_1_6"><i><font color="Green" title="E108">A568</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_21_1_3_1_1_6"><i><font color="Green" title="E109">A569</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_7"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_7"/><i><font color="Green" title="E108">A570</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_7"/><b>then </b><i><font color="Green" title="E109">A571</font></i>: 
<font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_3"><i><font color="Green" title="E88">A541</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_3"><i><font color="Green" title="E97">A550</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_3"><i><font color="Green" title="E98">A551</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_3"><i><font color="Green" title="E99">A552</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_3"><i><font color="Green" title="E105">A558</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_3"><i><font color="Green" title="E106">A559</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_3"><i><font color="Green" title="E107">A560</font></i></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></i>;<br/><a NAME="E3:33_1_1_21_1_3_1_1_7"/>
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> or <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_3"><i><font color="Green" title="E88">A541</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_3"><i><font color="Green" title="E97">A550</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_3"><i><font color="Green" title="E98">A551</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_3"><i><font color="Green" title="E99">A552</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_3"><i><font color="Green" title="E105">A558</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_3"><i><font color="Green" title="E106">A559</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_3"><i><font color="Green" title="E107">A560</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3_1_1_7"><i><font color="Green" title="E108">A570</font></i></a>, <a class="ref" href="goboard7.html#T25" title="GOBOARD7:th.25">GOBOARD7:25</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_3_1_1_7"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E62:33_1_1"><i><font color="Green" title="E76">A465</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21"><i><font color="Green" title="E85">A492</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3"><i><font color="Green" title="E86">A539</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_3"><i><font color="Green" title="E89">A542</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21_1_3"><i><font color="Green" title="E90">A543</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21_1_3"><i><font color="Green" title="E91">A544</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3_1_1_7"><i><font color="Green" title="E108">A570</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_21_1_3_1_1_7"><i><font color="Green" title="E109">A571</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_8"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_8"/><i><font color="Green" title="E108">A572</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_8"/><b>then </b>
( ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1</font> &amp; <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_3"><i><font color="Green" title="E88">A541</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_3"><i><font color="Green" title="E97">A550</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_3"><i><font color="Green" title="E98">A551</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_3"><i><font color="Green" title="E99">A552</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_3"><i><font color="Green" title="E105">A558</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_3"><i><font color="Green" title="E106">A559</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_3"><i><font color="Green" title="E107">A560</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_3_1_1_8"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E62:33_1_1"><i><font color="Green" title="E76">A465</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21"><i><font color="Green" title="E85">A492</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3"><i><font color="Green" title="E86">A539</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_3"><i><font color="Green" title="E89">A542</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21_1_3"><i><font color="Green" title="E90">A543</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21_1_3"><i><font color="Green" title="E91">A544</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3_1_1_8"><i><font color="Green" title="E108">A572</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_9"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_9"/><i><font color="Green" title="E108">A573</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_9"/><b>then </b>
( ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_3"><i><font color="Green" title="E88">A541</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_3"><i><font color="Green" title="E97">A550</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_3"><i><font color="Green" title="E98">A551</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_3"><i><font color="Green" title="E99">A552</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_3"><i><font color="Green" title="E105">A558</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_3"><i><font color="Green" title="E106">A559</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_3"><i><font color="Green" title="E107">A560</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_3_1_1_9"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E62:33_1_1"><i><font color="Green" title="E76">A465</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21"><i><font color="Green" title="E85">A492</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3"><i><font color="Green" title="E86">A539</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_3"><i><font color="Green" title="E89">A542</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21_1_3"><i><font color="Green" title="E90">A543</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21_1_3"><i><font color="Green" title="E91">A544</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3_1_1_9"><i><font color="Green" title="E108">A573</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_10"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_10"/><i><font color="Green" title="E108">A574</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_10"/><b>then </b><i><font color="Green" title="E109">A575</font></i>: 
<font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_3"><i><font color="Green" title="E88">A541</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_3"><i><font color="Green" title="E97">A550</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_3"><i><font color="Green" title="E98">A551</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_3"><i><font color="Green" title="E99">A552</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_3"><i><font color="Green" title="E105">A558</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_3"><i><font color="Green" title="E106">A559</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_3"><i><font color="Green" title="E107">A560</font></i></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></i>;<br/><a NAME="E3:33_1_1_21_1_3_1_1_10"/>
( <font color="Maroon" title="c20">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> or <font color="Maroon" title="c20">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_3"><i><font color="Green" title="E88">A541</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_3"><i><font color="Green" title="E97">A550</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_3"><i><font color="Green" title="E98">A551</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_3"><i><font color="Green" title="E99">A552</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_3"><i><font color="Green" title="E105">A558</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_3"><i><font color="Green" title="E106">A559</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_3"><i><font color="Green" title="E107">A560</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3_1_1_10"><i><font color="Green" title="E108">A574</font></i></a>, <a class="ref" href="goboard7.html#T25" title="GOBOARD7:th.25">GOBOARD7:25</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_3_1_1_10"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E61:33_1_1"><i><font color="Green" title="E75">A457</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3"><i><font color="Green" title="E86">A539</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_3"><i><font color="Green" title="E89">A542</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21_1_3"><i><font color="Green" title="E90">A543</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21_1_3"><i><font color="Green" title="E91">A544</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3_1_1_10"><i><font color="Green" title="E108">A574</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_21_1_3_1_1_10"><i><font color="Green" title="E109">A575</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_11"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_11"/><i><font color="Green" title="E108">A576</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_11"/><b>then </b><i><font color="Green" title="E109">A577</font></i>: 
<font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_3"><i><font color="Green" title="E88">A541</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_3"><i><font color="Green" title="E97">A550</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_3"><i><font color="Green" title="E98">A551</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_3"><i><font color="Green" title="E99">A552</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_3"><i><font color="Green" title="E105">A558</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_3"><i><font color="Green" title="E106">A559</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_3"><i><font color="Green" title="E107">A560</font></i></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></i>;<br/><a NAME="E3:33_1_1_21_1_3_1_1_11"/>
( <font color="Maroon" title="c20">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> or <font color="Maroon" title="c20">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_3"><i><font color="Green" title="E88">A541</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_3"><i><font color="Green" title="E97">A550</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_3"><i><font color="Green" title="E98">A551</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_3"><i><font color="Green" title="E99">A552</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_3"><i><font color="Green" title="E105">A558</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_3"><i><font color="Green" title="E106">A559</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_3"><i><font color="Green" title="E107">A560</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3_1_1_11"><i><font color="Green" title="E108">A576</font></i></a>, <a class="ref" href="goboard7.html#T25" title="GOBOARD7:th.25">GOBOARD7:25</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_3_1_1_11"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E62:33_1_1"><i><font color="Green" title="E76">A465</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21"><i><font color="Green" title="E85">A492</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3"><i><font color="Green" title="E86">A539</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_3"><i><font color="Green" title="E89">A542</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21_1_3"><i><font color="Green" title="E91">A544</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3_1_1_11"><i><font color="Green" title="E108">A576</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_21_1_3_1_1_11"><i><font color="Green" title="E109">A577</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_12"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_12"/><i><font color="Green" title="E108">A578</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_12"/><b>then </b>
( ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c25">j2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c25">j2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) or ( <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j1</font> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_3"><i><font color="Green" title="E88">A541</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_3"><i><font color="Green" title="E97">A550</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_3"><i><font color="Green" title="E98">A551</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_3"><i><font color="Green" title="E99">A552</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_3"><i><font color="Green" title="E105">A558</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_3"><i><font color="Green" title="E106">A559</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_3"><i><font color="Green" title="E107">A560</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_3_1_1_12"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E62:33_1_1"><i><font color="Green" title="E76">A465</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21"><i><font color="Green" title="E85">A492</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3"><i><font color="Green" title="E86">A539</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_3"><i><font color="Green" title="E89">A542</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21_1_3"><i><font color="Green" title="E90">A543</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21_1_3"><i><font color="Green" title="E91">A544</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3_1_1_12"><i><font color="Green" title="E108">A578</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_13"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_13"/><i><font color="Green" title="E108">A579</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_13"/><b>then </b><i><font color="Green" title="E109">A580</font></i>: 
<font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_3"><i><font color="Green" title="E88">A541</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_3"><i><font color="Green" title="E97">A550</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_3"><i><font color="Green" title="E98">A551</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_3"><i><font color="Green" title="E99">A552</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_3"><i><font color="Green" title="E105">A558</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_3"><i><font color="Green" title="E106">A559</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_3"><i><font color="Green" title="E107">A560</font></i></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></i>;<br/><a NAME="E3:33_1_1_21_1_3_1_1_13"/>
( <font color="Maroon" title="c21">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> or <font color="Maroon" title="c21">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_3"><i><font color="Green" title="E88">A541</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_3"><i><font color="Green" title="E97">A550</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_3"><i><font color="Green" title="E98">A551</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_3"><i><font color="Green" title="E99">A552</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_3"><i><font color="Green" title="E105">A558</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_3"><i><font color="Green" title="E106">A559</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_3"><i><font color="Green" title="E107">A560</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3_1_1_13"><i><font color="Green" title="E108">A579</font></i></a>, <a class="ref" href="goboard7.html#T24" title="GOBOARD7:th.24">GOBOARD7:24</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_3_1_1_13"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E61:33_1_1"><i><font color="Green" title="E75">A457</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3"><i><font color="Green" title="E86">A539</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_3"><i><font color="Green" title="E89">A542</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21_1_3"><i><font color="Green" title="E90">A543</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21_1_3"><i><font color="Green" title="E91">A544</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3_1_1_13"><i><font color="Green" title="E108">A579</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_21_1_3_1_1_13"><i><font color="Green" title="E109">A580</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_14"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_14"/><i><font color="Green" title="E108">A581</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_14"/><b>then </b>
( ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> &amp; <font color="Maroon" title="c21">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> &amp; <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_3"><i><font color="Green" title="E88">A541</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_3"><i><font color="Green" title="E97">A550</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_3"><i><font color="Green" title="E98">A551</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_3"><i><font color="Green" title="E99">A552</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_3"><i><font color="Green" title="E105">A558</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_3"><i><font color="Green" title="E106">A559</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_3"><i><font color="Green" title="E107">A560</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_3_1_1_14"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E62:33_1_1"><i><font color="Green" title="E76">A465</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21"><i><font color="Green" title="E85">A492</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3"><i><font color="Green" title="E86">A539</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_3"><i><font color="Green" title="E89">A542</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21_1_3"><i><font color="Green" title="E90">A543</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21_1_3"><i><font color="Green" title="E91">A544</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3_1_1_14"><i><font color="Green" title="E108">A581</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_15"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_15"/><i><font color="Green" title="E108">A582</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_15"/><b>then </b>
( ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c21">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) or ( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1'</font> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_3"><i><font color="Green" title="E88">A541</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_3"><i><font color="Green" title="E97">A550</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_3"><i><font color="Green" title="E98">A551</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_3"><i><font color="Green" title="E99">A552</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_3"><i><font color="Green" title="E105">A558</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_3"><i><font color="Green" title="E106">A559</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_3"><i><font color="Green" title="E107">A560</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_3_1_1_15"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E60:33_1_1"><i><font color="Green" title="E74">A434</font></i></a>, <a class="txt" href="jordan9.html#E62:33_1_1"><i><font color="Green" title="E76">A465</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21"><i><font color="Green" title="E84">A491</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21"><i><font color="Green" title="E85">A492</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3"><i><font color="Green" title="E86">A539</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_3"><i><font color="Green" title="E89">A542</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_21_1_3"><i><font color="Green" title="E90">A543</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21_1_3"><i><font color="Green" title="E91">A544</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3_1_1_15"><i><font color="Green" title="E108">A582</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_16"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_16"/><i><font color="Green" title="E108">A583</font></i>: 
( <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font> &amp; <font color="Maroon" title="c19">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c22">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2'</font> &amp; <font color="Maroon" title="c23">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_16"/><b>then </b><i><font color="Green" title="E109">A584</font></i>: 
<font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i1'</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_3"><i><font color="Green" title="E88">A541</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_3"><i><font color="Green" title="E97">A550</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_3"><i><font color="Green" title="E98">A551</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_3"><i><font color="Green" title="E99">A552</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_3"><i><font color="Green" title="E105">A558</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_3"><i><font color="Green" title="E106">A559</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_3"><i><font color="Green" title="E107">A560</font></i></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></i>;<br/><a NAME="E3:33_1_1_21_1_3_1_1_16"/>
( <font color="Maroon" title="c21">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> or <font color="Maroon" title="c21">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c21">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_21_1_3"><i><font color="Green" title="E88">A541</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_21_1_3"><i><font color="Green" title="E97">A550</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_21_1_3"><i><font color="Green" title="E98">A551</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_21_1_3"><i><font color="Green" title="E99">A552</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E22:33_1_1_21_1_3"><i><font color="Green" title="E105">A558</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_21_1_3"><i><font color="Green" title="E106">A559</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_21_1_3"><i><font color="Green" title="E107">A560</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3_1_1_16"><i><font color="Green" title="E108">A583</font></i></a>, <a class="ref" href="goboard7.html#T24" title="GOBOARD7:th.24">GOBOARD7:24</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_3_1_1_16"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E62:33_1_1"><i><font color="Green" title="E76">A465</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21"><i><font color="Green" title="E79">A486</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21"><i><font color="Green" title="E82">A489</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_21"><i><font color="Green" title="E85">A492</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3"><i><font color="Green" title="E86">A539</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_21_1_3"><i><font color="Green" title="E89">A542</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_21_1_3"><i><font color="Green" title="E91">A544</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_21_1_3"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_21_1_3"><i><font color="Green" title="E95">A548</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_21_1_3"><i><font color="Green" title="E101">A554</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_21_1_3"><i><font color="Green" title="E103">A556</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_21_1_3_1_1_16"><i><font color="Green" title="E108">A583</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_21_1_3_1_1_16"><i><font color="Green" title="E109">A584</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E26:33_1_1_21_1_3"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E65:33_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c15">g</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E47:33_1_1"><i><font color="Green" title="E67">A427</font></i></a>, <a class="txt" href="jordan9.html#E51:33_1_1"><i><font color="Green" title="E69">A429</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E66:33_1_1"/><b>then </b><i><font color="Green" title="E79">A585</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c11">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="c15">g</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E67:33_1_1"/><b>then </b><i><font color="Green" title="E80">A586</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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="c15">g</font>
 
;<br/>
<a NAME="E68:33_1_1"/>
not <font color="Maroon" title="c15">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="33_1_1_22"><b>proof </b></a><div class="add">

<b>take </b>
1
; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a><br/>

<b>take </b>
2
;<br/>

<b>thus </b><a NAME="E1:33_1_1_22"/><i><font color="Green" title="E81">A587</font></i>: 
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="c15">g</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>

<b>thus </b><a NAME="E2:33_1_1_22"/><i><font color="Green" title="E82">A588</font></i>: 
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="c15">g</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E66:33_1_1"><i><font color="Green" title="E79">A585</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>

<a NAME="E3:33_1_1_22"/><b>then </b>
<font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E59:33_1_1"><i><font color="Green" title="E73">A433</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_22"><i><font color="Green" title="E81">A587</font></i></a>, <a class="ref" href="goboard7.html#T31" title="GOBOARD7:th.31">GOBOARD7:31</a></i>;<br/>
<a NAME="E4:33_1_1_22"/><b>then </b>
<font color="Maroon" title="c15">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="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_22"><i><font color="Green" title="E81">A587</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:33_1_1_22"/>
<font color="Maroon" title="c15">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="c15">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_22"><i><font color="Green" title="E82">A588</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c16">g</font> = <font color="Maroon" title="c15">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> <b>by </b><i><a class="txt" href="jordan9.html#E38:33_1_1"><i><font color="Green" title="E61">A421</font></i></a>, <a class="txt" href="jordan9.html#E55:33_1_1"><i><font color="Green" title="E70">A430</font></i></a>, <a class="txt" href="jordan9.html#E57:33_1_1"><i><font color="Green" title="E71">A431</font></i></a>, <a class="txt" href="jordan9.html#E58:33_1_1"><i><font color="Green" title="E72">A432</font></i></a>, <a class="txt" href="jordan9.html#E64:33_1_1"><i><font color="Green" title="E78">A485</font></i></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></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_23"/>
<font color="Maroon" title="c11">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><a NAME="E2:33_1_1_23"/><b>then </b><i><font color="Green" title="E81">A589</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">m</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:33_1_1_23"/><i><font color="Green" title="E82">A590</font></i>: 
(  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c16">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#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp;  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c16">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#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 <b>by </b><i><a class="ref" href="goboard9.html#D1" title="GOBOARD9:def.1">GOBOARD9:def 1</a>, <a class="ref" href="goboard9.html#D2" title="GOBOARD9:def.2">GOBOARD9:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">Lg'</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">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="E5:33_1_1_23"/><i><font color="Green" title="E83">A591</font></i>: 
<font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">Lg'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_23_1"/><i><font color="Green" title="E84">A592</font></i>: 
<font color="Maroon" title="c18">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E2:33_1_1_23_1"/><i><font color="Green" title="E85">A593</font></i>: 
not <font color="Maroon" title="c18">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">Lg'</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c19">c</font> = <font color="Maroon" title="c18">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> <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_1"><i><font color="Green" title="E84">A592</font></i></a></i>;<br/>
<a NAME="E4:33_1_1_23_1"/>
<font color="Maroon" title="c19">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">g</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_23_1"><i><font color="Green" title="E85">A593</font></i></a>, <a class="ref" href="subset_1.html#T50" title="SUBSET_1:th.50">SUBSET_1:50</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">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> <b> such that </b><br/><a NAME="E6:33_1_1_23_1"/><i><font color="Green" title="E86">A594</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">i</font> &amp; <font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 <b>and </b><br/><a NAME="E7:33_1_1_23_1"/><i><font color="Green" title="E87">A595</font></i>: 
<font color="Maroon" title="c19">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c20">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sppol_2.html#T14" title="SPPOL_2:th.14">SPPOL_2:14</a></i>;<br/>
<a NAME="E8:33_1_1_23_1"/>
( <font color="Maroon" title="c20">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> &amp; <font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1_23_1"><i><font color="Green" title="E86">A594</font></i></a>, <a class="ref" href="goboard2.html#T3" title="GOBOARD2:th.3">GOBOARD2:3</a></i>;<br/>
<a NAME="E9:33_1_1_23_1"/><b>then </b><i><font color="Green" title="E88">A596</font></i>: 
( <font color="Maroon" title="c16">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c16">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>;<br/>
<a NAME="E10:33_1_1_23_1"/><i><font color="Green" title="E89">A597</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1_23_1"><i><font color="Green" title="E86">A594</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E11:33_1_1_23_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
;<br/>
<a NAME="E12:33_1_1_23_1"/><b>then </b><i><font color="Green" title="E90">A598</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1_23_1"><i><font color="Green" title="E86">A594</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E13:33_1_1_23_1"/><b>then </b>
<font color="Maroon" title="c19">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E51:33_1_1"><i><font color="Green" title="E69">A429</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_1"><i><font color="Green" title="E87">A595</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_1"><i><font color="Green" title="E88">A596</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_23_1"><i><font color="Green" title="E89">A597</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E14:33_1_1_23_1"/><b>then </b>
<font color="Maroon" title="c19">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="c13">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c13">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E51:33_1_1"><i><font color="Green" title="E69">A429</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_23_1"><i><font color="Green" title="E89">A597</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_23_1"><i><font color="Green" title="E90">A598</font></i></a>, <a class="ref" href="gobrd13.html#T30" title="GOBRD13:th.30">GOBRD13:30</a></i>;<br/>
<a NAME="E15:33_1_1_23_1"/><b>then </b>
<font color="Maroon" title="c19">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="c13">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E16:33_1_1_23_1"/><b>then </b>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c13">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_1"><i><font color="Green" title="E84">A592</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E17:33_1_1_23_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a>, <a class="txt" href="jordan9.html#E51:33_1_1"><i><font color="Green" title="E69">A429</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_23_1"><i><font color="Green" title="E89">A597</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_23_1"><i><font color="Green" title="E90">A598</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:33_1_1_23"/><i><font color="Green" title="E84">A599</font></i>: 
 <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#D8" title="EUCLID:def.8">EUCLID:def 8</a></i>;<br/><a NAME="E7:33_1_1_23"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">g</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="sppol_1.html#T49" title="SPPOL_1:th.49">SPPOL_1:49</a></i>;<br/><a NAME="E8:33_1_1_23"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">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>
 <b>by </b><i><a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></i>;<br/><a NAME="E9:33_1_1_23"/><b>then </b><i><font color="Green" title="E85">A600</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">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#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></i>;<br/><a NAME="E10:33_1_1_23"/><i><font color="Green" title="E86">A601</font></i>: 
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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_2"><b>proof </b></a><div class="add">

<b>let </b><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> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_23_2"/><i><font color="Green" title="E87">A602</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">j</font>
 <b>and </b><br/><a NAME="E2:33_1_1_23_2"/><i><font color="Green" title="E88">A603</font></i>: 
<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>
 ;<br/>

<a NAME="E3:33_1_1_23_2"/><i><font color="Green" title="E89">A604</font></i>: 
<span class="p1">(<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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E24:33_1_1"><i><font color="Green" title="E51">A351</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_2"><i><font color="Green" title="E88">A603</font></i></a></i>;<br/>
<a NAME="E4:33_1_1_23_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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_2"><i><font color="Green" title="E88">A603</font></i></a></i>;<br/>
<a NAME="E5:33_1_1_23_2"/><b>then </b>
 <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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E6:33_1_1_23_2"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_2"><i><font color="Green" title="E87">A602</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E7:33_1_1_23_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_4.html#K4" title="FINSEQ_4:func.4">/.</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">j</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_23_2"><i><font color="Green" title="E89">A604</font></i></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:33_1_1_23"/><i><font color="Green" title="E87">A605</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c13">f</font>
 <b>by </b><i><a class="ref" href="jordan3.html#T75" title="JORDAN3:th.75">JORDAN3:75</a></i>;<br/><a NAME="E12:33_1_1_23"/><i><font color="Green" title="E88">A606</font></i>: 
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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 holds <br/> not <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_3"><b>proof </b></a><div class="add">

<b>let </b><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> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_23_3"/><i><font color="Green" title="E89">A607</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">n</font>
 <b>and </b><br/><a NAME="E2:33_1_1_23_3"/><i><font color="Green" title="E90">A608</font></i>: 
<font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 ;<br/>

<a NAME="E3:33_1_1_23_3"/><i><font color="Green" title="E91">A609</font></i>: 
<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="c13">f</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E50:33_1_1"><i><font color="Green" title="E68">A428</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_3"><i><font color="Green" title="E90">A608</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c19">p</font> = <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">n</font>;<br/>
<b>assume </b><a NAME="E4:33_1_1_23_3"/>
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">g</font>
 ;<br/>

<b>then consider </b><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> <b> such that </b><br/><a NAME="E6:33_1_1_23_3"/><i><font color="Green" title="E92">A610</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">j</font>
 <b>and </b><br/><a NAME="E7:33_1_1_23_3"/><i><font color="Green" title="E93">A611</font></i>: 
<font color="Maroon" title="c20">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c13">f</font>
 <b>and </b><br/><a NAME="E8:33_1_1_23_3"/><i><font color="Green" title="E94">A612</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">f</font>,<font color="Maroon" title="c20">j</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E50:33_1_1"><i><font color="Green" title="E68">A428</font></i></a>, <a class="ref" href="jordan9.html#T9" target="_self" title="JORDAN9:th.9">Th9</a></i>;<br/>
<a NAME="E9:33_1_1_23_3"/><i><font color="Green" title="E95">A613</font></i>: 
<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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_23_3"><i><font color="Green" title="E90">A608</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E10:33_1_1_23_3"/><b>then </b><i><font color="Green" title="E96">A614</font></i>: 
<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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1_23_3"><i><font color="Green" title="E92">A610</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:33_1_1_23_3"/><i><font color="Green" title="E97">A615</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">m</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/>
<a NAME="E12:33_1_1_23_3"/><b>then </b><i><font color="Green" title="E98">A616</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">j</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_23"><i><font color="Green" title="E81">A589</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_3"><i><font color="Green" title="E92">A610</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:33_1_1_23_3"/><i><font color="Green" title="E99">A617</font></i>: 
<font color="Maroon" title="c20">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_3"><i><font color="Green" title="E93">A611</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E14:33_1_1_23_3"/><i><font color="Green" title="E100">A618</font></i>: 
( 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> &amp; 2 <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> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E15:33_1_1_23_3"/><i><font color="Green" title="E101">A619</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_3"><i><font color="Green" title="E89">A607</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_3"><i><font color="Green" title="E91">A609</font></i></a>, <a class="ref" href="jordan9.html#T8" target="_self" title="JORDAN9:th.8">Th8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_23_3_1"/>
( <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c20">j</font> or <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_3"><i><font color="Green" title="E93">A611</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_3"><i><font color="Green" title="E94">A612</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_23_3"><i><font color="Green" title="E98">A616</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_23_3"><i><font color="Green" title="E100">A618</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_23_3"><i><font color="Green" title="E101">A619</font></i></a>, <a class="ref" href="jordan9.html#T25" target="_self" title="JORDAN9:th.25">Th25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_3_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_23_3_1_1"/><i><font color="Green" title="E102">A620</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c20">j</font>
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_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>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_23_3"><i><font color="Green" title="E96">A614</font></i></a></i>;<br/><a NAME="E3:33_1_1_23_3_1_1"/><b>then </b><i><font color="Green" title="E103">A621</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_3"><i><font color="Green" title="E89">A607</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:33_1_1_23_3_1_1"/><i><font color="Green" title="E104">A622</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_3"><i><font color="Green" title="E92">A610</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_3"><i><font color="Green" title="E95">A613</font></i></a></i>;<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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">n</font>
<b>by </b><i><a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_3"><i><font color="Green" title="E89">A607</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_23_3"><i><font color="Green" title="E96">A614</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">n</font>
<b>by </b><i><a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_3"><i><font color="Green" title="E89">A607</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_3"><i><font color="Green" title="E91">A609</font></i></a></i>
<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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c20">j</font>
<b>by </b><i><a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_23_3"><i><font color="Green" title="E98">A616</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_23_3"><i><font color="Green" title="E99">A617</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_3_1_1"><i><font color="Green" title="E102">A620</font></i></a></i>
<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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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>
<b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a></i>
;<br/><b>hence </b><a NAME="E6:33_1_1_23_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E34:33_1_1"><i><font color="Green" title="E58">A418</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_23_3"><i><font color="Green" title="E98">A616</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_23_3"><i><font color="Green" title="E99">A617</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_3_1_1"><i><font color="Green" title="E103">A621</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_3_1_1"><i><font color="Green" title="E104">A622</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_3_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_23_3_1_2"/><i><font color="Green" title="E102">A623</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_3_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_3_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_23_3_1_2_1_1"/>
( <font color="Maroon" title="c20">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> or <font color="Maroon" title="c20">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font> )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_3"><i><font color="Green" title="E93">A611</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_3_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_23_3_1_2_1_1_1"/><i><font color="Green" title="E103">A624</font></i>: 
<font color="Maroon" title="c20">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font>
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_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="c11">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_3"><i><font color="Green" title="E95">A613</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_23_3"><i><font color="Green" title="E97">A615</font></i></a></i>;<br/><a NAME="E3:33_1_1_23_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E104">A625</font></i>: 
<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="c11">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_3"><i><font color="Green" title="E89">A607</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:33_1_1_23_3_1_2_1_1_1"/><i><font color="Green" title="E105">A626</font></i>: 
<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="c11">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_3"><i><font color="Green" title="E95">A613</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_23_3"><i><font color="Green" title="E97">A615</font></i></a></i>;<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">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">n</font>
<b>by </b><i><a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_3"><i><font color="Green" title="E89">A607</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_3"><i><font color="Green" title="E95">A613</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_23_3"><i><font color="Green" title="E97">A615</font></i></a></i>
<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_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_3"><i><font color="Green" title="E89">A607</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_3"><i><font color="Green" title="E91">A609</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_3_1_2"><i><font color="Green" title="E102">A623</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_3_1_2_1_1_1"><i><font color="Green" title="E103">A624</font></i></a></i>
<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">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">m</font>
<b>by </b><i><a class="txt" href="jordan9.html#E38:33_1_1"><i><font color="Green" title="E61">A421</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a></i>
<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">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a></i>
;<br/><b>hence </b><a NAME="E6:33_1_1_23_3_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E34:33_1_1"><i><font color="Green" title="E58">A418</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E43:33_1_1"><i><font color="Green" title="E65">A425</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_3_1_2_1_1_1"><i><font color="Green" title="E104">A625</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_3_1_2_1_1_1"><i><font color="Green" title="E105">A626</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_3_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_23_3_1_2_1_1_2"/><i><font color="Green" title="E103">A627</font></i>: 
<font color="Maroon" title="c20">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c21">l</font> = <font color="Maroon" title="c20">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><a NAME="E2:33_1_1_23_3_1_2_1_1_2"/><i><font color="Green" title="E104">A628</font></i>: 
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#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E3:33_1_1_23_3_1_2_1_1_2"/><i><font color="Green" title="E105">A629</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c18">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E10:33_1_1_23_3"><i><font color="Green" title="E96">A614</font></i></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></i>;<br/><a NAME="E4:33_1_1_23_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E106">A630</font></i>: 
<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#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:33_1_1_23_3_1_2_1_1_2"/><b>then </b>
<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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a></i>;<br/><a NAME="E6:33_1_1_23_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E107">A631</font></i>: 
<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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_3"><i><font color="Green" title="E89">A607</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:33_1_1_23_3_1_2_1_1_2"/><i><font color="Green" title="E108">A632</font></i>: 
<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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_3_1_2_1_1_2"><i><font color="Green" title="E105">A629</font></i></a></i>;<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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">n</font>
<b>by </b><i><a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_3"><i><font color="Green" title="E89">A607</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_3_1_2_1_1_2"><i><font color="Green" title="E106">A630</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">n</font>
<b>by </b><i><a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_3"><i><font color="Green" title="E89">A607</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_3"><i><font color="Green" title="E91">A609</font></i></a></i>
<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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_3_1_2"><i><font color="Green" title="E102">A623</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_3_1_2_1_1_2"><i><font color="Green" title="E103">A627</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_3_1_2_1_1_2"><i><font color="Green" title="E104">A628</font></i></a></i>
<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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a></i>
;<br/><b>hence </b><a NAME="E9:33_1_1_23_3_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E34:33_1_1"><i><font color="Green" title="E58">A418</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_3_1_2_1_1_2"><i><font color="Green" title="E103">A627</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_3_1_2_1_1_2"><i><font color="Green" title="E104">A628</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_3_1_2_1_1_2"><i><font color="Green" title="E107">A631</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_3_1_2_1_1_2"><i><font color="Green" title="E108">A632</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_23_3_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E13:33_1_1_23"/><i><font color="Green" title="E89">A633</font></i>: 
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#K5" title="TOPREAL1:func.5">L~</a> <font color="Olive" title="b1">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c13">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#K5" title="TOPREAL1:func.5">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#K1" title="NAT_1:func.1">+</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="c13">f</font> &amp; <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">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="33_1_1_23_4"><b>proof </b></a><div class="add">

<b>let </b><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>;<br/>

<b>assume </b><a NAME="E1:33_1_1_23_4"/><i><font color="Green" title="E90">A634</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c18">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c13">f</font>
 ;<br/>

<b>let </b><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>;<br/>

<b>assume </b><a NAME="E2:33_1_1_23_4"/><i><font color="Green" title="E91">A635</font></i>: 
<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#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c18">h</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ;<br/>

<b>let </b><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> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:33_1_1_23_4"/><i><font color="Green" title="E92">A636</font></i>: 
( 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#K1" title="NAT_1:func.1">+</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="c13">f</font> )
 <b>and </b><br/><a NAME="E4:33_1_1_23_4"/><i><font color="Green" title="E93">A637</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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>
 <b>and </b><br/><a NAME="E5:33_1_1_23_4"/><i><font color="Green" title="E94">A638</font></i>: 
not <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c18">h</font>
 ;<br/>

<b>reconsider </b><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="c13">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#K5" title="TOPREAL1:func.5">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="E7:33_1_1_23_4"/><i><font color="Green" title="E95">A639</font></i>: 
<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="c13">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>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E8:33_1_1_23_4"/><i><font color="Green" title="E96">A640</font></i>: 
<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="c13">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#K5" title="TOPREAL1:func.5">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>
 
<b>by </b><i><a class="ref" href="subset_1.html#T32" title="SUBSET_1:th.32">SUBSET_1:32</a></i>;<br/>
<a NAME="E9:33_1_1_23_4"/><i><font color="Green" title="E97">A641</font></i>: 
<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="33_1_1_23_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:33_1_1_23_4_1"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c13">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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_4"><i><font color="Green" title="E92">A636</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c22">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:33_1_1_23_4_1"/><i><font color="Green" title="E98">A642</font></i>: 
( <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="c13">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="c22">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><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> <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_23_4_1"><i><font color="Green" title="E98">A642</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_4_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c24">a</font> = <font color="Maroon" title="c23">p</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_4_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_23_4_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#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c18">h</font>
 ;<br/><b>then consider </b><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> <b> such that </b><br/><a NAME="E3:33_1_1_23_4_1_1_1"/><i><font color="Green" title="E99">A643</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">j</font>
 <b>and </b><br/><a NAME="E4:33_1_1_23_4_1_1_1"/><i><font color="Green" title="E100">A644</font></i>: 
<font color="Maroon" title="c25">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c13">f</font>
 <b>and </b><br/><a NAME="E5:33_1_1_23_4_1_1_1"/><i><font color="Green" title="E101">A645</font></i>: 
<font color="Maroon" title="c23">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">f</font>,<font color="Maroon" title="c25">j</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_4"><i><font color="Green" title="E90">A634</font></i></a>, <a class="ref" href="sppol_2.html#T13" title="SPPOL_2:th.13">SPPOL_2:13</a></i>;<br/><a NAME="E6:33_1_1_23_4_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="c13">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="c13">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>
 <b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_4_1_1_1"><i><font color="Green" title="E99">A643</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_4_1_1_1"><i><font color="Green" title="E100">A644</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23_4_1_1_1"><i><font color="Green" title="E101">A645</font></i></a>, <a class="ref" href="gobrd13.html#T30" title="GOBRD13:th.30">GOBRD13:30</a></i>;<br/><a NAME="E7:33_1_1_23_4_1_1_1"/><b>then </b><i><font color="Green" title="E102">A646</font></i>: 
<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="c13">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>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:33_1_1_23_4_1_1_1"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c13">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>
 <b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_4_1_1_1"><i><font color="Green" title="E99">A643</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_4_1_1_1"><i><font color="Green" title="E100">A644</font></i></a></i>;<br/><b>hence </b><a NAME="E9:33_1_1_23_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_23_4_1"><i><font color="Green" title="E98">A642</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_4_1_1_1"><i><font color="Green" title="E102">A646</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:33_1_1_23_4_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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_23_4_1"><i><font color="Green" title="E98">A642</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>thus </b><a NAME="E3:33_1_1_23_4_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>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_23_4_1"><i><font color="Green" title="E98">A642</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:33_1_1_23_4_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>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:33_1_1_23_4"/><i><font color="Green" title="E98">A647</font></i>: 
 <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="c13">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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_4"><i><font color="Green" title="E92">A636</font></i></a>, <a class="ref" href="jordan9.html#T12" target="_self" title="JORDAN9:th.12">Th12</a></i>;<br/>
<a NAME="E11:33_1_1_23_4"/>
 <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="c13">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#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c13">f</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_4"><i><font color="Green" title="E92">A636</font></i></a>, <a class="ref" href="jordan9.html#T17" target="_self" title="JORDAN9:th.17">Th17</a></i>;<br/>
<a NAME="E12:33_1_1_23_4"/><b>then </b>
 <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="c13">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#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c18">h</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_4"><i><font color="Green" title="E90">A634</font></i></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E13:33_1_1_23_4"/><b>then </b><i><font color="Green" title="E99">A648</font></i>: 
 <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="c13">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#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c18">h</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T43" title="SUBSET_1:th.43">SUBSET_1:43</a></i>;<br/>
<a NAME="E14:33_1_1_23_4"/>
 <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="c13">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="c13">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>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></i>;<br/>
<a NAME="E15:33_1_1_23_4"/><b>then </b><i><font color="Green" title="E100">A649</font></i>: 
 <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="c13">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> <font color="Maroon" title="c21">rc</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E8:33_1_1_23_4"><i><font color="Green" title="E96">A640</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_23_4"><i><font color="Green" title="E99">A648</font></i></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E16:33_1_1_23_4"/>
<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="c13">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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_4"><i><font color="Green" title="E92">A636</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_4"><i><font color="Green" title="E95">A639</font></i></a>, <a class="ref" href="jordan9.html#T13" target="_self" title="JORDAN9:th.13">Th13</a></i>;<br/>
<a NAME="E17:33_1_1_23_4"/><b>then </b><i><font color="Green" title="E101">A650</font></i>: 
<font color="Maroon" title="c21">rc</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="jordan9.html#E10:33_1_1_23_4"><i><font color="Green" title="E98">A647</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_23_4"><i><font color="Green" title="E100">A649</font></i></a>, <a class="ref" href="connsp_1.html#T19" title="CONNSP_1:th.19">CONNSP_1:19</a></i>;<br/>
<a NAME="E18:33_1_1_23_4"/>
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="c13">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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_4"><i><font color="Green" title="E92">A636</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self" title="JORDAN9:th.10">Th10</a></i>;<br/>
<a NAME="E19:33_1_1_23_4"/><b>then </b>
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1_23_4"><i><font color="Green" title="E94">A638</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E20:33_1_1_23_4"/><b>then </b><i><font color="Green" title="E102">A651</font></i>: 
<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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_23_4"><i><font color="Green" title="E93">A637</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E21:33_1_1_23_4"/>
<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#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c18">h</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="txt" href="jordan9.html#E8:33_1_1_23_4"><i><font color="Green" title="E96">A640</font></i></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E22:33_1_1_23_4"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_23_4"><i><font color="Green" title="E91">A635</font></i></a>, <a class="txt" href="jordan9.html#E17:33_1_1_23_4"><i><font color="Green" title="E101">A650</font></i></a>, <a class="txt" href="jordan9.html#E20:33_1_1_23_4"><i><font color="Green" title="E102">A651</font></i></a>, <a class="ref" href="goboard9.html#T6" title="GOBOARD9:th.6">GOBOARD9:6</a></i>;<br/>
<b>hence </b><a NAME="E23:33_1_1_23_4"/>
<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>
 <b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_23_4"><i><font color="Green" title="E97">A641</font></i></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:33_1_1_23"/><i><font color="Green" title="E90">A652</font></i>: 
<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="c16">g</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_5"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c18">rg</font> =  <a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font>;<br/>
<a NAME="E1:33_1_1_23_5"/><i><font color="Green" title="E91">A653</font></i>: 
 <a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E58:33_1_1"><i><font color="Green" title="E72">A432</font></i></a>, <a class="ref" href="jordan9.html#T7" target="_self" title="JORDAN9:th.7">Th7</a></i>;<br/>
<a NAME="E2:33_1_1_23_5"/><i><font color="Green" title="E92">A654</font></i>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E66:33_1_1"><i><font color="Green" title="E79">A585</font></i></a>, <a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>;<br/>
<a NAME="E3:33_1_1_23_5"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a></i>;<br/>
<a NAME="E4:33_1_1_23_5"/><b>then </b><i><font color="Green" title="E93">A655</font></i>: 
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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/>
<a NAME="E5:33_1_1_23_5"/>
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="c16">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_23_5"><i><font color="Green" title="E92">A654</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E6:33_1_1_23_5"/><b>then </b><i><font color="Green" title="E94">A656</font></i>: 
<span class="p1">(<span class="default"><span 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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/>
<b>set </b><font color="Maroon" title="c19">p</font> = <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1;<br/>
<b>set </b><font color="Maroon" title="c20">q</font> = <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2;<br/>
<b>set </b><font color="Maroon" title="c21">a</font> = <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c22">l</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>;<br/>
<a NAME="E7:33_1_1_23_5"/><i><font color="Green" title="E95">A657</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">m</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E38:33_1_1"><i><font color="Green" title="E61">A421</font></i></a>, <a class="txt" href="jordan9.html#E57:33_1_1"><i><font color="Green" title="E71">A431</font></i></a>, <a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></i>;<br/>
<a NAME="E8:33_1_1_23_5"/><i><font color="Green" title="E96">A658</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">m</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/>
<a NAME="E9:33_1_1_23_5"/><b>then </b><i><font color="Green" title="E97">A659</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_23"><i><font color="Green" title="E81">A589</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E10:33_1_1_23_5"/>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan9.html#E66:33_1_1"><i><font color="Green" title="E79">A585</font></i></a>, <a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a></i>;<br/>
<a NAME="E11:33_1_1_23_5"/><b>then </b><i><font color="Green" title="E98">A660</font></i>: 
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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/>
<a NAME="E12:33_1_1_23_5"/><i><font color="Green" title="E99">A661</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="c16">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="jordan9.html#E66:33_1_1"><i><font color="Green" title="E79">A585</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E13:33_1_1_23_5"/><i><font color="Green" title="E100">A662</font></i>: 
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_23"><i><font color="Green" title="E81">A589</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
<b>by </b><i><a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="ref" href="binarith.html#T56" title="BINARITH:th.56">BINARITH:56</a></i>
<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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
<b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_23_5"><i><font color="Green" title="E98">A660</font></i></a>, <a class="ref" href="binarith.html#T55" title="BINARITH:th.55">BINARITH:55</a></i>
<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="c16">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<b>by </b><i><a class="txt" href="jordan9.html#E12:33_1_1_23_5"><i><font color="Green" title="E99">A661</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<b>by </b><i><a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E51:33_1_1"><i><font color="Green" title="E69">A429</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_23_5"><i><font color="Green" title="E100">A662</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c10">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1

;<br/>
<a NAME="E15:33_1_1_23_5"/><b>then </b><i><font color="Green" title="E101">A663</font></i>: 
<font color="Maroon" title="c10">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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
;<br/>
<a NAME="E16:33_1_1_23_5"/><b>then </b><i><font color="Green" title="E102">A664</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E17:33_1_1_23_5"/>
<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> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E18:33_1_1_23_5"/><b>then </b><i><font color="Green" title="E103">A665</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_23_5"><i><font color="Green" title="E98">A660</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E19:33_1_1_23_5"/>
<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> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
 
<b>by </b><i><a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/>
<a NAME="E20:33_1_1_23_5"/><b>then </b><i><font color="Green" title="E104">A666</font></i>: 
<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> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c16">g</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_23_5"><i><font color="Green" title="E98">A660</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E21:33_1_1_23_5"/>
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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E67:33_1_1"><i><font color="Green" title="E80">A586</font></i></a>, <a class="ref" href="sppol_1.html#T5" title="SPPOL_1:th.5">SPPOL_1:5</a></i>;<br/>
<a NAME="E22:33_1_1_23_5"/><b>then </b><i><font color="Green" title="E105">A667</font></i>: 
<span 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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<b>then </b><i><font color="Green" title="E106">A668</font></i>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2 = 
<font color="Maroon" title="c16">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4: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> <font color="Maroon" title="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1_23_5"><i><font color="Green" title="E104">A666</font></i></a>, <a class="ref" href="finseq_5.html#T69" title="FINSEQ_5:th.69">FINSEQ_5:69</a></i>
<br/>.= 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E20:33_1_1_23_5"><i><font color="Green" title="E104">A666</font></i></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>
;<br/>
<a NAME="E24:33_1_1_23_5"/><i><font color="Green" title="E107">A669</font></i>: 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c13">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a>, <a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a></i>;<br/>
<a NAME="E25:33_1_1_23_5"/><i><font color="Green" title="E108">A670</font></i>: 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c13">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_23_5"><i><font color="Green" title="E101">A663</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5"><i><font color="Green" title="E103">A665</font></i></a></i>;<br/>
<a NAME="E26:33_1_1_23_5"/>
not <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">g</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E12:33_1_1_23"><i><font color="Green" title="E88">A606</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a></i>;<br/>
<a NAME="E27:33_1_1_23_5"/><b>then </b><i><font color="Green" title="E109">A671</font></i>: 
not <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sppol_2.html#T22" title="SPPOL_2:th.22">SPPOL_2:22</a></i>;<br/>
<a NAME="E28:33_1_1_23_5"/><i><font color="Green" title="E110">A672</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<a NAME="E29:33_1_1_23_5"/><i><font color="Green" title="E111">A673</font></i>: 
<font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E30:33_1_1_23_5"/><b>then </b>
<font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a></i>;<br/>
<a NAME="E31:33_1_1_23_5"/><b>then </b><i><font color="Green" title="E112">A674</font></i>: 
<font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E32:33_1_1_23_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_23_5"><i><font color="Green" title="E98">A660</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E33:33_1_1_23_5"/><b>then </b>
<font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E34:33_1_1_23_5"/><b>then </b><i><font color="Green" title="E113">A675</font></i>: 
<font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a></i>;<br/>
<b>consider </b><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> <b> such that </b><br/><a NAME="E36:33_1_1_23_5"/><i><font color="Green" title="E114">A676</font></i>: 
<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>
 <b>and </b><br/><a NAME="E37:33_1_1_23_5"/><i><font color="Green" title="E115">A677</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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>
 <b>and </b><br/><a NAME="E38:33_1_1_23_5"/><i><font color="Green" title="E116">A678</font></i>: 
<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>
 <b>and </b><br/><a NAME="E39:33_1_1_23_5"/><i><font color="Green" title="E117">A679</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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>
 <b>and </b><br/><a NAME="E40:33_1_1_23_5"/><i><font color="Green" title="E118">A680</font></i>: 
( ( <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_5"><i><font color="Green" title="E91">A653</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_5"><i><font color="Green" title="E92">A654</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/>
<a NAME="E41:33_1_1_23_5"/><i><font color="Green" title="E119">A681</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">p1</font> &amp; <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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">p2</font> &amp; <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> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_23_5_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E40:33_1_1_23_5"><i><font color="Green" title="E118">A680</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_1"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_1"/><i><font color="Green" title="E120">A682</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q2</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c27">a1</font>, <font color="Maroon" title="c28">a2</font>, <font color="Maroon" title="c29">p'1</font>, <font color="Maroon" title="c30">p'2</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> <b> such that </b><br/><a NAME="E3:33_1_1_23_5_3_1"/><i><font color="Green" title="E121">A683</font></i>: 
<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>
 <b>and </b><br/><a NAME="E4:33_1_1_23_5_3_1"/><i><font color="Green" title="E122">A684</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>and </b><br/><a NAME="E5:33_1_1_23_5_3_1"/><i><font color="Green" title="E123">A685</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">p'1</font>,<font color="Maroon" title="c30">p'2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E6:33_1_1_23_5_3_1"/><i><font color="Green" title="E124">A686</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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">p'1</font>,<font color="Maroon" title="c30">p'2</font>
 <b>and </b><br/><a NAME="E7:33_1_1_23_5_3_1"/><i><font color="Green" title="E125">A687</font></i>: 
( ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">p'2</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E8:33_1_1_23_5_3_1"/><i><font color="Green" title="E126">A688</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">a1</font> &amp; <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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">a2</font> &amp; <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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_1"><i><font color="Green" title="E121">A683</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><b>thus </b><a NAME="E9:33_1_1_23_5_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="c16">g</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_5_3_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_23_5_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">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">p'2</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_23_5_3_1"><i><font color="Green" title="E125">A687</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_1_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">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_1_1_1_1"/><b>then </b><i><font color="Green" title="E127">A689</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p2</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23_5_3_1"><i><font color="Green" title="E123">A685</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5_3_1"><i><font color="Green" title="E124">A686</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_23_5_3_1_1_1_1"/><i><font color="Green" title="E128">A690</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c11">m</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a></i>;<br/><a NAME="E4:33_1_1_23_5_3_1_1_1_1"/><i><font color="Green" title="E129">A691</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c11">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a></i>;<br/><a NAME="E5:33_1_1_23_5_3_1_1_1_1"/><i><font color="Green" title="E130">A692</font></i>: 
<font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E6:33_1_1_23_5_3_1_1_1_1"/><i><font color="Green" title="E131">A693</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="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> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E24:33_1_1"><i><font color="Green" title="E51">A351</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E47:33_1_1"><i><font color="Green" title="E67">A427</font></i></a></i>;<br/><a NAME="E7:33_1_1_23_5_3_1_1_1_1"/><i><font color="Green" title="E132">A694</font></i>: 
( <font color="Maroon" title="c24">p2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c28">a2</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_23_5_3_1_1_1_1"><i><font color="Green" title="E127">A689</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><i><font color="Green" title="E133">A695</font></i>: <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E50:33_1_1"><i><font color="Green" title="E68">A428</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a></i>
<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">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23_5_3_1_1_1_1"><i><font color="Green" title="E130">A692</font></i></a></i>
;<br/><a NAME="E9:33_1_1_23_5_3_1_1_1_1"/><i><font color="Green" title="E134">A696</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c11">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">m</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a></i>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c13">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c24">p2</font>
<b>by </b><i><a class="txt" href="jordan9.html#E38:33_1_1"><i><font color="Green" title="E61">A421</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_23_5"><i><font color="Green" title="E101">A663</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5"><i><font color="Green" title="E103">A665</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_23_5"><i><font color="Green" title="E106">A668</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E38:33_1_1_23_5"><i><font color="Green" title="E116">A678</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1_23_5"><i><font color="Green" title="E117">A679</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_1"><i><font color="Green" title="E120">A682</font></i></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></i>
<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="c11">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_1"><i><font color="Green" title="E121">A683</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_1"><i><font color="Green" title="E122">A684</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_5_3_1_1_1_1"><i><font color="Green" title="E127">A689</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_1_1_1_1"><i><font color="Green" title="E128">A690</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_1_1_1_1"><i><font color="Green" title="E129">A691</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5_3_1_1_1_1"><i><font color="Green" title="E133">A695</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5_3_1_1_1_1"><i><font color="Green" title="E134">A696</font></i></a>, <a class="ref" href="gobrd13.html#T35" title="GOBRD13:th.35">GOBRD13:35</a></i>
;<br/><a NAME="E11:33_1_1_23_5_3_1_1_1_1"/><b>then </b>
<font 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="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E25:33_1_1"><i><font color="Green" title="E52">A352</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23"><i><font color="Green" title="E81">A589</font></i></a>, <a class="txt" href="jordan9.html#E25:33_1_1_23_5"><i><font color="Green" title="E108">A670</font></i></a></i>;<br/><a NAME="E12:33_1_1_23_5_3_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c13">f</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E47:33_1_1"><i><font color="Green" title="E67">A427</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E28:33_1_1_23_5"><i><font color="Green" title="E110">A672</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5_3_1_1_1_1"><i><font color="Green" title="E131">A693</font></i></a>, <a class="ref" href="gobrd13.html#T45" title="GOBRD13:th.45">GOBRD13:45</a></i>;<br/><a NAME="E13:33_1_1_23_5_3_1_1_1_1"/><b>then </b><i><font color="Green" title="E135">A697</font></i>: 
( <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c24">p2</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="c23">p1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E28:33_1_1_23_5"><i><font color="Green" title="E110">A672</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_1"><i><font color="Green" title="E121">A683</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_1"><i><font color="Green" title="E122">A684</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5_3_1_1_1_1"><i><font color="Green" title="E132">A694</font></i></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></i>;<br/><a NAME="E14:33_1_1_23_5_3_1_1_1_1"/><b>then </b><i><font color="Green" title="E136">A698</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">p1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <font color="Maroon" title="c23">p1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><b>set </b><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="c16">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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span></span>)</span>;<br/><a NAME="E15:33_1_1_23_5_3_1_1_1_1"/><i><font color="Green" title="E137">A699</font></i>: 
<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="c16">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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">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="c16">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_5"><i><font color="Green" title="E91">A653</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5"><i><font color="Green" title="E93">A655</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5"><i><font color="Green" title="E94">A656</font></i></a>, <a class="ref" href="jordan9.html#T29" target="_self" title="JORDAN9:th.29">Th29</a></i>;<br/><a NAME="E16:33_1_1_23_5_3_1_1_1_1"/><i><font color="Green" title="E138">A700</font></i>: 
( <font color="Maroon" title="c24">p2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">a2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c23">p1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p1</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1_23_5"><i><font color="Green" title="E119">A681</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_5_3_1_1_1_1"><i><font color="Green" title="E127">A689</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/><a NAME="E17:33_1_1_23_5_3_1_1_1_1"/><i><font color="Green" title="E139">A701</font></i>: 
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>
 <b>by </b><i><a class="txt" href="jordan9.html#E66:33_1_1"><i><font color="Green" title="E79">A585</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E18:33_1_1_23_5_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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E22:33_1_1_23_5"><i><font color="Green" title="E105">A667</font></i></a>, <a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E140">A702</font></i>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> = 
<font color="Maroon" title="c16">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan9.html#E17:33_1_1_23_5_3_1_1_1_1"><i><font color="Green" title="E139">A701</font></i></a>, <a class="ref" href="finseq_5.html#T69" title="FINSEQ_5:th.69">FINSEQ_5:69</a></i>
<br/>.= 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E28:33_1_1_23_5"><i><font color="Green" title="E110">A672</font></i></a>, <a class="txt" href="jordan9.html#E17:33_1_1_23_5_3_1_1_1_1"><i><font color="Green" title="E139">A701</font></i></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>
;<br/><span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<font color="Maroon" title="c16">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
<b>by </b><i><a class="txt" href="jordan9.html#E38:33_1_1"><i><font color="Green" title="E61">A421</font></i></a>, <a class="txt" href="jordan9.html#E55:33_1_1"><i><font color="Green" title="E70">A430</font></i></a>, <a class="txt" href="jordan9.html#E57:33_1_1"><i><font color="Green" title="E71">A431</font></i></a>, <a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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>
<b>by </b><i><a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="c16">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>
;<br/><a NAME="E21:33_1_1_23_5_3_1_1_1_1"/><b>then </b>
 <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="c16">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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c28">a2</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_5"><i><font color="Green" title="E91">A653</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5"><i><font color="Green" title="E93">A655</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5"><i><font color="Green" title="E94">A656</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_23_5_3_1_1_1_1"><i><font color="Green" title="E135">A697</font></i></a>, <a class="txt" href="jordan9.html#E16:33_1_1_23_5_3_1_1_1_1"><i><font color="Green" title="E138">A700</font></i></a>, <a class="txt" href="jordan9.html#E19:33_1_1_23_5_3_1_1_1_1"><i><font color="Green" title="E140">A702</font></i></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></i>;<br/><a NAME="E22:33_1_1_23_5_3_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c16">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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1_23_5"><i><font color="Green" title="E119">A681</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_1"><i><font color="Green" title="E122">A684</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5_3_1"><i><font color="Green" title="E126">A688</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_5_3_1_1_1_1"><i><font color="Green" title="E127">A689</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_23_5_3_1_1_1_1"><i><font color="Green" title="E136">A698</font></i></a>, <a class="txt" href="jordan9.html#E16:33_1_1_23_5_3_1_1_1_1"><i><font color="Green" title="E138">A700</font></i></a>, <a class="ref" href="jordan9.html#T22" target="_self" title="JORDAN9:th.22">Th22</a></i>;<br/><a NAME="E23:33_1_1_23_5_3_1_1_1_1"/><b>then </b><i><font color="Green" title="E141">A703</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c16">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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E27:33_1_1_23_5"><i><font color="Green" title="E109">A671</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E24:33_1_1_23_5_3_1_1_1_1"/><i><font color="Green" title="E142">A704</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c13">f</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_23"><i><font color="Green" title="E87">A605</font></i></a>, <a class="ref" href="sppol_2.html#T22" title="SPPOL_2:th.22">SPPOL_2:22</a></i>;<br/><a NAME="E25:33_1_1_23_5_3_1_1_1_1"/>
 <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="c16">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#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="goboard9.html#D2" title="GOBOARD9:def.2">GOBOARD9:def 2</a></i>;<br/><b>hence </b><a NAME="E26:33_1_1_23_5_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="c16">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_23"><i><font color="Green" title="E89">A633</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_23_5"><i><font color="Green" title="E109">A671</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_23_5_3_1_1_1_1"><i><font color="Green" title="E137">A699</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_23_5_3_1_1_1_1"><i><font color="Green" title="E141">A703</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_23_5_3_1_1_1_1"><i><font color="Green" title="E142">A704</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_1_1_1_2"/><i><font color="Green" title="E127">A705</font></i>: 
( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_1_1_1_2"/><b>then </b><i><font color="Green" title="E128">A706</font></i>: 
( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23_5_3_1"><i><font color="Green" title="E123">A685</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5_3_1"><i><font color="Green" title="E124">A686</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_23_5_3_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c25">q1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">a1</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_1"><i><font color="Green" title="E120">A682</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/><b>then </b> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c13">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
<b>by </b><i><a class="txt" href="jordan9.html#E38:33_1_1"><i><font color="Green" title="E61">A421</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_23_5"><i><font color="Green" title="E101">A663</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5"><i><font color="Green" title="E103">A665</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_23_5"><i><font color="Green" title="E106">A668</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E38:33_1_1_23_5"><i><font color="Green" title="E116">A678</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1_23_5"><i><font color="Green" title="E117">A679</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_1"><i><font color="Green" title="E120">A682</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_5_3_1_1_1_2"><i><font color="Green" title="E128">A706</font></i></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></i>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c13">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_1"><i><font color="Green" title="E121">A683</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_1"><i><font color="Green" title="E122">A684</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23_5_3_1"><i><font color="Green" title="E123">A685</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5_3_1"><i><font color="Green" title="E124">A686</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_1_1_1_2"><i><font color="Green" title="E127">A705</font></i></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></i>
;<br/><b>hence </b><a NAME="E5:33_1_1_23_5_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="c16">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a>, <a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E25:33_1_1_23_5"><i><font color="Green" title="E108">A670</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_1_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_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">p'1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_1_1_1_3"/><b>then </b>
( <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#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23_5_3_1"><i><font color="Green" title="E123">A685</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5_3_1"><i><font color="Green" title="E124">A686</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>then </b> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c13">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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>
<b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_1"><i><font color="Green" title="E121">A683</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_1"><i><font color="Green" title="E122">A684</font></i></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></i>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c13">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
<b>by </b><i><a class="txt" href="jordan9.html#E38:33_1_1"><i><font color="Green" title="E61">A421</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_23_5"><i><font color="Green" title="E101">A663</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5"><i><font color="Green" title="E103">A665</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_23_5"><i><font color="Green" title="E106">A668</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E38:33_1_1_23_5"><i><font color="Green" title="E116">A678</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1_23_5"><i><font color="Green" title="E117">A679</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_1"><i><font color="Green" title="E120">A682</font></i></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></i>
;<br/><b>hence </b><a NAME="E4:33_1_1_23_5_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="c16">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_23_5"><i><font color="Green" title="E101">A663</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5"><i><font color="Green" title="E103">A665</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_23_5"><i><font color="Green" title="E107">A669</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_1_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_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">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_1_1_1_4"/><b>then </b><i><font color="Green" title="E127">A707</font></i>: 
( <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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_1"><i><font color="Green" title="E120">A682</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23_5_3_1"><i><font color="Green" title="E123">A685</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5_3_1"><i><font color="Green" title="E124">A686</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E29:33_1_1_23_5"><i><font color="Green" title="E111">A673</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E50:33_1_1"><i><font color="Green" title="E68">A428</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1_23_5"><i><font color="Green" title="E117">A679</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_1"><i><font color="Green" title="E122">A684</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_5_3_1_1_1_4"><i><font color="Green" title="E127">A707</font></i></a></i>
<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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E16:33_1_1_23_5"><i><font color="Green" title="E102">A664</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5"><i><font color="Green" title="E103">A665</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_23_5"><i><font color="Green" title="E106">A668</font></i></a></i>
<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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:33_1_1_23_5_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="c16">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E34:33_1_1"><i><font color="Green" title="E58">A418</font></i></a>, <a class="txt" href="jordan9.html#E16:33_1_1_23_5"><i><font color="Green" title="E102">A664</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5"><i><font color="Green" title="E103">A665</font></i></a>, <a class="txt" href="jordan9.html#E31:33_1_1_23_5"><i><font color="Green" title="E112">A674</font></i></a>, <a class="txt" href="jordan9.html#E34:33_1_1_23_5"><i><font color="Green" title="E113">A675</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_2"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_2"/><i><font color="Green" title="E120">A708</font></i>: 
( <font color="Maroon" title="c23">p1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c27">a1</font>, <font color="Maroon" title="c28">a2</font>, <font color="Maroon" title="c29">p'1</font>, <font color="Maroon" title="c30">p'2</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> <b> such that </b><br/><a NAME="E3:33_1_1_23_5_3_2"/><i><font color="Green" title="E121">A709</font></i>: 
<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>
 <b>and </b><br/><a NAME="E4:33_1_1_23_5_3_2"/><i><font color="Green" title="E122">A710</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>and </b><br/><a NAME="E5:33_1_1_23_5_3_2"/><i><font color="Green" title="E123">A711</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">p'1</font>,<font color="Maroon" title="c30">p'2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E6:33_1_1_23_5_3_2"/><i><font color="Green" title="E124">A712</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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">p'1</font>,<font color="Maroon" title="c30">p'2</font>
 <b>and </b><br/><a NAME="E7:33_1_1_23_5_3_2"/><i><font color="Green" title="E125">A713</font></i>: 
( ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">p'2</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E8:33_1_1_23_5_3_2"/><i><font color="Green" title="E126">A714</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">a1</font> &amp; <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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">a2</font> &amp; <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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_2"><i><font color="Green" title="E121">A709</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><b>thus </b><a NAME="E9:33_1_1_23_5_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="c16">g</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_5_3_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_23_5_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">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">p'2</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_23_5_3_2"><i><font color="Green" title="E125">A713</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_2_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_2_1_1_1"/><i><font color="Green" title="E127">A715</font></i>: 
( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_2_1_1_1"/><b>then </b><i><font color="Green" title="E128">A716</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p2</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23_5_3_2"><i><font color="Green" title="E123">A711</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5_3_2"><i><font color="Green" title="E124">A712</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_23_5_3_2_1_1_1"/><b>then </b><i><font color="Green" title="E129">A717</font></i>: 
<font color="Maroon" title="c26">q2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">a2</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_2"><i><font color="Green" title="E120">A708</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c13">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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>
<b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_2"><i><font color="Green" title="E121">A709</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_2"><i><font color="Green" title="E122">A710</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23_5_3_2"><i><font color="Green" title="E123">A711</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5_3_2"><i><font color="Green" title="E124">A712</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_2_1_1_1"><i><font color="Green" title="E127">A715</font></i></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></i>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c13">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
<b>by </b><i><a class="txt" href="jordan9.html#E38:33_1_1"><i><font color="Green" title="E61">A421</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_23_5"><i><font color="Green" title="E101">A663</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5"><i><font color="Green" title="E103">A665</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_23_5"><i><font color="Green" title="E106">A668</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E38:33_1_1_23_5"><i><font color="Green" title="E116">A678</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1_23_5"><i><font color="Green" title="E117">A679</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_2"><i><font color="Green" title="E120">A708</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_5_3_2_1_1_1"><i><font color="Green" title="E128">A716</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_2_1_1_1"><i><font color="Green" title="E129">A717</font></i></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></i>
;<br/><b>hence </b><a NAME="E5:33_1_1_23_5_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="c16">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_23_5"><i><font color="Green" title="E101">A663</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5"><i><font color="Green" title="E103">A665</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_23_5"><i><font color="Green" title="E107">A669</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_2_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_2_1_1_2"/>
( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_2_1_1_2"/><b>then </b><i><font color="Green" title="E127">A718</font></i>: 
( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23_5_3_2"><i><font color="Green" title="E123">A711</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5_3_2"><i><font color="Green" title="E124">A712</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_23_5_3_2_1_1_2"/><i><font color="Green" title="E128">A719</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c11">m</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a></i>;<br/><a NAME="E4:33_1_1_23_5_3_2_1_1_2"/><i><font color="Green" title="E129">A720</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c11">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a></i>;<br/><a NAME="E5:33_1_1_23_5_3_2_1_1_2"/><i><font color="Green" title="E130">A721</font></i>: 
<font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E6:33_1_1_23_5_3_2_1_1_2"/><i><font color="Green" title="E131">A722</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="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> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E24:33_1_1"><i><font color="Green" title="E51">A351</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E47:33_1_1"><i><font color="Green" title="E67">A427</font></i></a></i>;<br/><a NAME="E7:33_1_1_23_5_3_2_1_1_2"/><i><font color="Green" title="E132">A723</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><i><font color="Green" title="E133">A724</font></i>: <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E50:33_1_1"><i><font color="Green" title="E68">A428</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a></i>
<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">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23_5_3_2_1_1_2"><i><font color="Green" title="E130">A721</font></i></a></i>
;<br/><a NAME="E9:33_1_1_23_5_3_2_1_1_2"/><i><font color="Green" title="E134">A725</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c11">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">m</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a></i>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c13">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
<b>by </b><i><a class="txt" href="jordan9.html#E38:33_1_1"><i><font color="Green" title="E61">A421</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_23_5"><i><font color="Green" title="E101">A663</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5"><i><font color="Green" title="E103">A665</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_23_5"><i><font color="Green" title="E106">A668</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E38:33_1_1_23_5"><i><font color="Green" title="E116">A678</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1_23_5"><i><font color="Green" title="E117">A679</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_2"><i><font color="Green" title="E120">A708</font></i></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></i>
<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="c11">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_2"><i><font color="Green" title="E121">A709</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_2"><i><font color="Green" title="E122">A710</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_5_3_2_1_1_2"><i><font color="Green" title="E127">A718</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_2_1_1_2"><i><font color="Green" title="E128">A719</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_2_1_1_2"><i><font color="Green" title="E129">A720</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5_3_2_1_1_2"><i><font color="Green" title="E133">A724</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5_3_2_1_1_2"><i><font color="Green" title="E134">A725</font></i></a>, <a class="ref" href="gobrd13.html#T37" title="GOBRD13:th.37">GOBRD13:37</a></i>
;<br/><a NAME="E11:33_1_1_23_5_3_2_1_1_2"/><b>then </b>
<font 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="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E25:33_1_1"><i><font color="Green" title="E52">A352</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23"><i><font color="Green" title="E81">A589</font></i></a>, <a class="txt" href="jordan9.html#E25:33_1_1_23_5"><i><font color="Green" title="E108">A670</font></i></a></i>;<br/><a NAME="E12:33_1_1_23_5_3_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c13">f</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E47:33_1_1"><i><font color="Green" title="E67">A427</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E28:33_1_1_23_5"><i><font color="Green" title="E110">A672</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5_3_2_1_1_2"><i><font color="Green" title="E131">A722</font></i></a>, <a class="ref" href="gobrd13.html#T45" title="GOBRD13:th.45">GOBRD13:45</a></i>;<br/><a NAME="E13:33_1_1_23_5_3_2_1_1_2"/><b>then </b><i><font color="Green" title="E135">A726</font></i>: 
( <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <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#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E28:33_1_1_23_5"><i><font color="Green" title="E110">A672</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_2"><i><font color="Green" title="E121">A709</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_2"><i><font color="Green" title="E122">A710</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_5_3_2_1_1_2"><i><font color="Green" title="E127">A718</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5_3_2_1_1_2"><i><font color="Green" title="E132">A723</font></i></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></i>;<br/><a NAME="E14:33_1_1_23_5_3_2_1_1_2"/><b>then </b><i><font color="Green" title="E136">A727</font></i>: 
<font color="Maroon" title="c24">p2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><b>set </b><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="c16">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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span></span>)</span>;<br/><a NAME="E15:33_1_1_23_5_3_2_1_1_2"/><i><font color="Green" title="E137">A728</font></i>: 
<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="c16">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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">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="c16">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_5"><i><font color="Green" title="E91">A653</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5"><i><font color="Green" title="E93">A655</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5"><i><font color="Green" title="E94">A656</font></i></a>, <a class="ref" href="jordan9.html#T29" target="_self" title="JORDAN9:th.29">Th29</a></i>;<br/><a NAME="E16:33_1_1_23_5_3_2_1_1_2"/><i><font color="Green" title="E138">A729</font></i>: 
<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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_23_5_3_2_1_1_2"><i><font color="Green" title="E127">A718</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/><a NAME="E17:33_1_1_23_5_3_2_1_1_2"/><i><font color="Green" title="E139">A730</font></i>: 
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>
 <b>by </b><i><a class="txt" href="jordan9.html#E66:33_1_1"><i><font color="Green" title="E79">A585</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E18:33_1_1_23_5_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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E22:33_1_1_23_5"><i><font color="Green" title="E105">A667</font></i></a>, <a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E140">A731</font></i>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> = 
<font color="Maroon" title="c16">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan9.html#E17:33_1_1_23_5_3_2_1_1_2"><i><font color="Green" title="E139">A730</font></i></a>, <a class="ref" href="finseq_5.html#T69" title="FINSEQ_5:th.69">FINSEQ_5:69</a></i>
<br/>.= 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E28:33_1_1_23_5"><i><font color="Green" title="E110">A672</font></i></a>, <a class="txt" href="jordan9.html#E17:33_1_1_23_5_3_2_1_1_2"><i><font color="Green" title="E139">A730</font></i></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>
;<br/><span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<font color="Maroon" title="c16">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
<b>by </b><i><a class="txt" href="jordan9.html#E38:33_1_1"><i><font color="Green" title="E61">A421</font></i></a>, <a class="txt" href="jordan9.html#E55:33_1_1"><i><font color="Green" title="E70">A430</font></i></a>, <a class="txt" href="jordan9.html#E57:33_1_1"><i><font color="Green" title="E71">A431</font></i></a>, <a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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>
<b>by </b><i><a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="c16">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>
;<br/><a NAME="E21:33_1_1_23_5_3_2_1_1_2"/><b>then </b>
 <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="c16">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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c28">a2</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_5"><i><font color="Green" title="E91">A653</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5"><i><font color="Green" title="E93">A655</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5"><i><font color="Green" title="E94">A656</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_5_3_2_1_1_2"><i><font color="Green" title="E127">A718</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_23_5_3_2_1_1_2"><i><font color="Green" title="E135">A726</font></i></a>, <a class="txt" href="jordan9.html#E19:33_1_1_23_5_3_2_1_1_2"><i><font color="Green" title="E140">A731</font></i></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></i>;<br/><a NAME="E22:33_1_1_23_5_3_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c16">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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1_23_5"><i><font color="Green" title="E119">A681</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_2"><i><font color="Green" title="E122">A710</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5_3_2"><i><font color="Green" title="E126">A714</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_5_3_2_1_1_2"><i><font color="Green" title="E127">A718</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_23_5_3_2_1_1_2"><i><font color="Green" title="E136">A727</font></i></a>, <a class="txt" href="jordan9.html#E16:33_1_1_23_5_3_2_1_1_2"><i><font color="Green" title="E138">A729</font></i></a>, <a class="ref" href="jordan9.html#T22" target="_self" title="JORDAN9:th.22">Th22</a></i>;<br/><a NAME="E23:33_1_1_23_5_3_2_1_1_2"/><b>then </b><i><font color="Green" title="E141">A732</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c16">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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E27:33_1_1_23_5"><i><font color="Green" title="E109">A671</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E24:33_1_1_23_5_3_2_1_1_2"/><i><font color="Green" title="E142">A733</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c13">f</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_23"><i><font color="Green" title="E87">A605</font></i></a>, <a class="ref" href="sppol_2.html#T22" title="SPPOL_2:th.22">SPPOL_2:22</a></i>;<br/><a NAME="E25:33_1_1_23_5_3_2_1_1_2"/>
 <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="c16">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#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="goboard9.html#D2" title="GOBOARD9:def.2">GOBOARD9:def 2</a></i>;<br/><b>hence </b><a NAME="E26:33_1_1_23_5_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="c16">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_23"><i><font color="Green" title="E89">A633</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_23_5"><i><font color="Green" title="E109">A671</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_23_5_3_2_1_1_2"><i><font color="Green" title="E137">A728</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_23_5_3_2_1_1_2"><i><font color="Green" title="E141">A732</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_23_5_3_2_1_1_2"><i><font color="Green" title="E142">A733</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_2_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_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">p'1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_2_1_1_3"/><b>then </b><i><font color="Green" title="E127">A734</font></i>: 
( <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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_2"><i><font color="Green" title="E120">A708</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23_5_3_2"><i><font color="Green" title="E123">A711</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5_3_2"><i><font color="Green" title="E124">A712</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E29:33_1_1_23_5"><i><font color="Green" title="E111">A673</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E50:33_1_1"><i><font color="Green" title="E68">A428</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1_23_5"><i><font color="Green" title="E117">A679</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_2"><i><font color="Green" title="E122">A710</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_5_3_2_1_1_3"><i><font color="Green" title="E127">A734</font></i></a></i>
<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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E16:33_1_1_23_5"><i><font color="Green" title="E102">A664</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5"><i><font color="Green" title="E103">A665</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_23_5"><i><font color="Green" title="E106">A668</font></i></a></i>
<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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:33_1_1_23_5_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="c16">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E34:33_1_1"><i><font color="Green" title="E58">A418</font></i></a>, <a class="txt" href="jordan9.html#E16:33_1_1_23_5"><i><font color="Green" title="E102">A664</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5"><i><font color="Green" title="E103">A665</font></i></a>, <a class="txt" href="jordan9.html#E31:33_1_1_23_5"><i><font color="Green" title="E112">A674</font></i></a>, <a class="txt" href="jordan9.html#E34:33_1_1_23_5"><i><font color="Green" title="E113">A675</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_2_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_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">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_2_1_1_4"/><b>then </b><i><font color="Green" title="E127">A735</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23_5_3_2"><i><font color="Green" title="E123">A711</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5_3_2"><i><font color="Green" title="E124">A712</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c13">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
<b>by </b><i><a class="txt" href="jordan9.html#E38:33_1_1"><i><font color="Green" title="E61">A421</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_23_5"><i><font color="Green" title="E101">A663</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5"><i><font color="Green" title="E103">A665</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_23_5"><i><font color="Green" title="E106">A668</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E38:33_1_1_23_5"><i><font color="Green" title="E116">A678</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1_23_5"><i><font color="Green" title="E117">A679</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_2"><i><font color="Green" title="E120">A708</font></i></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></i>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c13">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_2"><i><font color="Green" title="E121">A709</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_2"><i><font color="Green" title="E122">A710</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_5_3_2_1_1_4"><i><font color="Green" title="E127">A735</font></i></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></i>
;<br/><b>hence </b><a NAME="E4:33_1_1_23_5_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="c16">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a>, <a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E25:33_1_1_23_5"><i><font color="Green" title="E108">A670</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_3"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_3"/><i><font color="Green" title="E120">A736</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c27">a1</font>, <font color="Maroon" title="c28">a2</font>, <font color="Maroon" title="c29">p'1</font>, <font color="Maroon" title="c30">p'2</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> <b> such that </b><br/><a NAME="E3:33_1_1_23_5_3_3"/><i><font color="Green" title="E121">A737</font></i>: 
<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>
 <b>and </b><br/><a NAME="E4:33_1_1_23_5_3_3"/><i><font color="Green" title="E122">A738</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>and </b><br/><a NAME="E5:33_1_1_23_5_3_3"/><i><font color="Green" title="E123">A739</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">p'1</font>,<font color="Maroon" title="c30">p'2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E6:33_1_1_23_5_3_3"/><i><font color="Green" title="E124">A740</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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">p'1</font>,<font color="Maroon" title="c30">p'2</font>
 <b>and </b><br/><a NAME="E7:33_1_1_23_5_3_3"/><i><font color="Green" title="E125">A741</font></i>: 
( ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">p'2</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E8:33_1_1_23_5_3_3"/><i><font color="Green" title="E126">A742</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">a1</font> &amp; <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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">a2</font> &amp; <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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_3"><i><font color="Green" title="E121">A737</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><b>thus </b><a NAME="E9:33_1_1_23_5_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="c16">g</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_5_3_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_23_5_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">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">p'2</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_23_5_3_3"><i><font color="Green" title="E125">A741</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_3_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_3_1_1_1"/><i><font color="Green" title="E127">A743</font></i>: 
( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_3_1_1_1"/><b>then </b><i><font color="Green" title="E128">A744</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p2</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23_5_3_3"><i><font color="Green" title="E123">A739</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5_3_3"><i><font color="Green" title="E124">A740</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_23_5_3_3_1_1_1"/><b>then </b><i><font color="Green" title="E129">A745</font></i>: 
<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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_3"><i><font color="Green" title="E120">A736</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/><a NAME="E4:33_1_1_23_5_3_3_1_1_1"/><i><font color="Green" title="E130">A746</font></i>: 
<font color="Maroon" title="c26">q2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">a2</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_3"><i><font color="Green" title="E120">A736</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_5_3_3_1_1_1"><i><font color="Green" title="E128">A744</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c13">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E38:33_1_1"><i><font color="Green" title="E61">A421</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_23_5"><i><font color="Green" title="E101">A663</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5"><i><font color="Green" title="E103">A665</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_23_5"><i><font color="Green" title="E106">A668</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E38:33_1_1_23_5"><i><font color="Green" title="E116">A678</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1_23_5"><i><font color="Green" title="E117">A679</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_3"><i><font color="Green" title="E120">A736</font></i></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></i>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c13">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_3"><i><font color="Green" title="E121">A737</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_3"><i><font color="Green" title="E122">A738</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23_5_3_3"><i><font color="Green" title="E123">A739</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5_3_3"><i><font color="Green" title="E124">A740</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_3_1_1_1"><i><font color="Green" title="E127">A743</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_3_1_1_1"><i><font color="Green" title="E129">A745</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_3_1_1_1"><i><font color="Green" title="E130">A746</font></i></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></i>
;<br/><b>hence </b><a NAME="E6:33_1_1_23_5_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="c16">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a>, <a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E25:33_1_1_23_5"><i><font color="Green" title="E108">A670</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_3_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_3_1_1_2"/>
( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_3_1_1_2"/><b>then </b><i><font color="Green" title="E127">A747</font></i>: 
( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23_5_3_3"><i><font color="Green" title="E123">A739</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5_3_3"><i><font color="Green" title="E124">A740</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E29:33_1_1_23_5"><i><font color="Green" title="E111">A673</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E50:33_1_1"><i><font color="Green" title="E68">A428</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1_23_5"><i><font color="Green" title="E117">A679</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_3"><i><font color="Green" title="E120">A736</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_3"><i><font color="Green" title="E122">A738</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_5_3_3_1_1_2"><i><font color="Green" title="E127">A747</font></i></a></i>
<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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E16:33_1_1_23_5"><i><font color="Green" title="E102">A664</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5"><i><font color="Green" title="E103">A665</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_23_5"><i><font color="Green" title="E106">A668</font></i></a></i>
<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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:33_1_1_23_5_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="c16">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E34:33_1_1"><i><font color="Green" title="E58">A418</font></i></a>, <a class="txt" href="jordan9.html#E16:33_1_1_23_5"><i><font color="Green" title="E102">A664</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5"><i><font color="Green" title="E103">A665</font></i></a>, <a class="txt" href="jordan9.html#E31:33_1_1_23_5"><i><font color="Green" title="E112">A674</font></i></a>, <a class="txt" href="jordan9.html#E34:33_1_1_23_5"><i><font color="Green" title="E113">A675</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_3_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_3_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">p'1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_3_1_1_3"/><b>then </b><i><font color="Green" title="E127">A748</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23_5_3_3"><i><font color="Green" title="E123">A739</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5_3_3"><i><font color="Green" title="E124">A740</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_23_5_3_3_1_1_3"/><i><font color="Green" title="E128">A749</font></i>: 
<font color="Maroon" title="c23">p1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q1</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_3"><i><font color="Green" title="E120">A736</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/><a NAME="E4:33_1_1_23_5_3_3_1_1_3"/><i><font color="Green" title="E129">A750</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c11">m</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a></i>;<br/><a NAME="E5:33_1_1_23_5_3_3_1_1_3"/><i><font color="Green" title="E130">A751</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c11">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a></i>;<br/><a NAME="E6:33_1_1_23_5_3_3_1_1_3"/><i><font color="Green" title="E131">A752</font></i>: 
<font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E7:33_1_1_23_5_3_3_1_1_3"/><i><font color="Green" title="E132">A753</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="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> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E24:33_1_1"><i><font color="Green" title="E51">A351</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E47:33_1_1"><i><font color="Green" title="E67">A427</font></i></a></i>;<br/><a NAME="E8:33_1_1_23_5_3_3_1_1_3"/>
<font color="Maroon" title="c23">p1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c23">p1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E9:33_1_1_23_5_3_3_1_1_3"/><b>then </b><i><font color="Green" title="E133">A754</font></i>: 
( <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c27">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c23">p1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">p2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_23_5_3_3_1_1_3"><i><font color="Green" title="E127">A748</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><i><font color="Green" title="E134">A755</font></i>: <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E50:33_1_1"><i><font color="Green" title="E68">A428</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a></i>
<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">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5_3_3_1_1_3"><i><font color="Green" title="E131">A752</font></i></a></i>
;<br/><a NAME="E11:33_1_1_23_5_3_3_1_1_3"/><i><font color="Green" title="E135">A756</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c11">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">m</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a></i>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c13">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E38:33_1_1"><i><font color="Green" title="E61">A421</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_23_5"><i><font color="Green" title="E101">A663</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5"><i><font color="Green" title="E103">A665</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_23_5"><i><font color="Green" title="E106">A668</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E38:33_1_1_23_5"><i><font color="Green" title="E116">A678</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1_23_5"><i><font color="Green" title="E117">A679</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_3"><i><font color="Green" title="E120">A736</font></i></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></i>
<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="c11">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_3"><i><font color="Green" title="E120">A736</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_3"><i><font color="Green" title="E121">A737</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_3"><i><font color="Green" title="E122">A738</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_5_3_3_1_1_3"><i><font color="Green" title="E127">A748</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_3_1_1_3"><i><font color="Green" title="E128">A749</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_3_1_1_3"><i><font color="Green" title="E129">A750</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23_5_3_3_1_1_3"><i><font color="Green" title="E130">A751</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_23_5_3_3_1_1_3"><i><font color="Green" title="E134">A755</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_1_23_5_3_3_1_1_3"><i><font color="Green" title="E135">A756</font></i></a>, <a class="ref" href="gobrd13.html#T39" title="GOBRD13:th.39">GOBRD13:39</a></i>
;<br/><a NAME="E13:33_1_1_23_5_3_3_1_1_3"/><b>then </b>
<font 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="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E25:33_1_1"><i><font color="Green" title="E52">A352</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23"><i><font color="Green" title="E81">A589</font></i></a>, <a class="txt" href="jordan9.html#E25:33_1_1_23_5"><i><font color="Green" title="E108">A670</font></i></a></i>;<br/><a NAME="E14:33_1_1_23_5_3_3_1_1_3"/><b>then </b>
<font color="Maroon" title="c13">f</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E47:33_1_1"><i><font color="Green" title="E67">A427</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E28:33_1_1_23_5"><i><font color="Green" title="E110">A672</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5_3_3_1_1_3"><i><font color="Green" title="E132">A753</font></i></a>, <a class="ref" href="gobrd13.html#T45" title="GOBRD13:th.45">GOBRD13:45</a></i>;<br/><a NAME="E15:33_1_1_23_5_3_3_1_1_3"/><b>then </b><i><font color="Green" title="E136">A757</font></i>: 
( <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp; <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="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E28:33_1_1_23_5"><i><font color="Green" title="E110">A672</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_3"><i><font color="Green" title="E121">A737</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_3"><i><font color="Green" title="E122">A738</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5_3_3_1_1_3"><i><font color="Green" title="E133">A754</font></i></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></i>;<br/><a NAME="E16:33_1_1_23_5_3_3_1_1_3"/><b>then </b><i><font color="Green" title="E137">A758</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">p2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <font color="Maroon" title="c24">p2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><b>set </b><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="c16">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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span></span>)</span>;<br/><a NAME="E17:33_1_1_23_5_3_3_1_1_3"/><i><font color="Green" title="E138">A759</font></i>: 
<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="c16">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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">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="c16">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_5"><i><font color="Green" title="E91">A653</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5"><i><font color="Green" title="E93">A655</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5"><i><font color="Green" title="E94">A656</font></i></a>, <a class="ref" href="jordan9.html#T29" target="_self" title="JORDAN9:th.29">Th29</a></i>;<br/><a NAME="E18:33_1_1_23_5_3_3_1_1_3"/><i><font color="Green" title="E139">A760</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c24">p2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p2</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1_23_5"><i><font color="Green" title="E119">A681</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><a NAME="E19:33_1_1_23_5_3_3_1_1_3"/><i><font color="Green" title="E140">A761</font></i>: 
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>
 <b>by </b><i><a class="txt" href="jordan9.html#E66:33_1_1"><i><font color="Green" title="E79">A585</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E20:33_1_1_23_5_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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E22:33_1_1_23_5"><i><font color="Green" title="E105">A667</font></i></a>, <a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E141">A762</font></i>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> = 
<font color="Maroon" title="c16">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1_23_5_3_3_1_1_3"><i><font color="Green" title="E140">A761</font></i></a>, <a class="ref" href="finseq_5.html#T69" title="FINSEQ_5:th.69">FINSEQ_5:69</a></i>
<br/>.= 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E28:33_1_1_23_5"><i><font color="Green" title="E110">A672</font></i></a>, <a class="txt" href="jordan9.html#E19:33_1_1_23_5_3_3_1_1_3"><i><font color="Green" title="E140">A761</font></i></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>
;<br/><span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<font color="Maroon" title="c16">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
<b>by </b><i><a class="txt" href="jordan9.html#E38:33_1_1"><i><font color="Green" title="E61">A421</font></i></a>, <a class="txt" href="jordan9.html#E55:33_1_1"><i><font color="Green" title="E70">A430</font></i></a>, <a class="txt" href="jordan9.html#E57:33_1_1"><i><font color="Green" title="E71">A431</font></i></a>, <a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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>
<b>by </b><i><a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="c16">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>
;<br/><a NAME="E23:33_1_1_23_5_3_3_1_1_3"/><b>then </b>
 <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="c16">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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_5"><i><font color="Green" title="E91">A653</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5"><i><font color="Green" title="E93">A655</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5"><i><font color="Green" title="E94">A656</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_23_5_3_3_1_1_3"><i><font color="Green" title="E136">A757</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5_3_3_1_1_3"><i><font color="Green" title="E139">A760</font></i></a>, <a class="txt" href="jordan9.html#E21:33_1_1_23_5_3_3_1_1_3"><i><font color="Green" title="E141">A762</font></i></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></i>;<br/><a NAME="E24:33_1_1_23_5_3_3_1_1_3"/><b>then </b>
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c16">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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1_23_5"><i><font color="Green" title="E119">A681</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_3"><i><font color="Green" title="E122">A738</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5_3_3"><i><font color="Green" title="E126">A742</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_5_3_3_1_1_3"><i><font color="Green" title="E127">A748</font></i></a>, <a class="txt" href="jordan9.html#E16:33_1_1_23_5_3_3_1_1_3"><i><font color="Green" title="E137">A758</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5_3_3_1_1_3"><i><font color="Green" title="E139">A760</font></i></a>, <a class="ref" href="jordan9.html#T22" target="_self" title="JORDAN9:th.22">Th22</a></i>;<br/><a NAME="E25:33_1_1_23_5_3_3_1_1_3"/><b>then </b><i><font color="Green" title="E142">A763</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c16">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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E27:33_1_1_23_5"><i><font color="Green" title="E109">A671</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E26:33_1_1_23_5_3_3_1_1_3"/><i><font color="Green" title="E143">A764</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c13">f</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_23"><i><font color="Green" title="E87">A605</font></i></a>, <a class="ref" href="sppol_2.html#T22" title="SPPOL_2:th.22">SPPOL_2:22</a></i>;<br/><a NAME="E27:33_1_1_23_5_3_3_1_1_3"/>
 <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="c16">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#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="goboard9.html#D2" title="GOBOARD9:def.2">GOBOARD9:def 2</a></i>;<br/><b>hence </b><a NAME="E28:33_1_1_23_5_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="c16">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_23"><i><font color="Green" title="E89">A633</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_23_5"><i><font color="Green" title="E109">A671</font></i></a>, <a class="txt" href="jordan9.html#E17:33_1_1_23_5_3_3_1_1_3"><i><font color="Green" title="E138">A759</font></i></a>, <a class="txt" href="jordan9.html#E25:33_1_1_23_5_3_3_1_1_3"><i><font color="Green" title="E142">A763</font></i></a>, <a class="txt" href="jordan9.html#E26:33_1_1_23_5_3_3_1_1_3"><i><font color="Green" title="E143">A764</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_3_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_3_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">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_3_1_1_4"/><b>then </b><i><font color="Green" title="E127">A765</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23_5_3_3"><i><font color="Green" title="E123">A739</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5_3_3"><i><font color="Green" title="E124">A740</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_23_5_3_3_1_1_4"/><b>then </b>
<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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_3"><i><font color="Green" title="E120">A736</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/><b>then </b> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c13">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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>
<b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_3"><i><font color="Green" title="E120">A736</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_3"><i><font color="Green" title="E121">A737</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_3"><i><font color="Green" title="E122">A738</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_5_3_3_1_1_4"><i><font color="Green" title="E127">A765</font></i></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></i>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c13">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
<b>by </b><i><a class="txt" href="jordan9.html#E38:33_1_1"><i><font color="Green" title="E61">A421</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_23_5"><i><font color="Green" title="E101">A663</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5"><i><font color="Green" title="E103">A665</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_23_5"><i><font color="Green" title="E106">A668</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E38:33_1_1_23_5"><i><font color="Green" title="E116">A678</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1_23_5"><i><font color="Green" title="E117">A679</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_3"><i><font color="Green" title="E120">A736</font></i></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></i>
;<br/><b>hence </b><a NAME="E5:33_1_1_23_5_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="c16">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_23_5"><i><font color="Green" title="E101">A663</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5"><i><font color="Green" title="E103">A665</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_23_5"><i><font color="Green" title="E107">A669</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_4"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_4"/><i><font color="Green" title="E120">A766</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c27">a1</font>, <font color="Maroon" title="c28">a2</font>, <font color="Maroon" title="c29">p'1</font>, <font color="Maroon" title="c30">p'2</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> <b> such that </b><br/><a NAME="E3:33_1_1_23_5_3_4"/><i><font color="Green" title="E121">A767</font></i>: 
<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>
 <b>and </b><br/><a NAME="E4:33_1_1_23_5_3_4"/><i><font color="Green" title="E122">A768</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>and </b><br/><a NAME="E5:33_1_1_23_5_3_4"/><i><font color="Green" title="E123">A769</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">p'1</font>,<font color="Maroon" title="c30">p'2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E6:33_1_1_23_5_3_4"/><i><font color="Green" title="E124">A770</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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">p'1</font>,<font color="Maroon" title="c30">p'2</font>
 <b>and </b><br/><a NAME="E7:33_1_1_23_5_3_4"/><i><font color="Green" title="E125">A771</font></i>: 
( ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">p'2</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E8:33_1_1_23_5_3_4"/><i><font color="Green" title="E126">A772</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">a1</font> &amp; <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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">a2</font> &amp; <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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_4"><i><font color="Green" title="E121">A767</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><b>thus </b><a NAME="E9:33_1_1_23_5_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="c16">g</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_5_3_4_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_23_5_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">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">p'2</font> ) or ( <font color="Maroon" title="c27">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_23_5_3_4"><i><font color="Green" title="E125">A771</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_4_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_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">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_4_1_1_1"/><b>then </b><i><font color="Green" title="E127">A773</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p2</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23_5_3_4"><i><font color="Green" title="E123">A769</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5_3_4"><i><font color="Green" title="E124">A770</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E29:33_1_1_23_5"><i><font color="Green" title="E111">A673</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E50:33_1_1"><i><font color="Green" title="E68">A428</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1_23_5"><i><font color="Green" title="E117">A679</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_4"><i><font color="Green" title="E120">A766</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_4"><i><font color="Green" title="E122">A768</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_5_3_4_1_1_1"><i><font color="Green" title="E127">A773</font></i></a></i>
<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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E16:33_1_1_23_5"><i><font color="Green" title="E102">A664</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5"><i><font color="Green" title="E103">A665</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_23_5"><i><font color="Green" title="E106">A668</font></i></a></i>
<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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:33_1_1_23_5_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="c16">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E34:33_1_1"><i><font color="Green" title="E58">A418</font></i></a>, <a class="txt" href="jordan9.html#E16:33_1_1_23_5"><i><font color="Green" title="E102">A664</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5"><i><font color="Green" title="E103">A665</font></i></a>, <a class="txt" href="jordan9.html#E31:33_1_1_23_5"><i><font color="Green" title="E112">A674</font></i></a>, <a class="txt" href="jordan9.html#E34:33_1_1_23_5"><i><font color="Green" title="E113">A675</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_4_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_4_1_1_2"/><i><font color="Green" title="E127">A774</font></i>: 
( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_4_1_1_2"/><b>then </b><i><font color="Green" title="E128">A775</font></i>: 
( <font color="Maroon" title="c27">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23_5_3_4"><i><font color="Green" title="E123">A769</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5_3_4"><i><font color="Green" title="E124">A770</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_23_5_3_4_1_1_2"/><b>then </b><i><font color="Green" title="E129">A776</font></i>: 
<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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_4"><i><font color="Green" title="E120">A766</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/><a NAME="E4:33_1_1_23_5_3_4_1_1_2"/><i><font color="Green" title="E130">A777</font></i>: 
<font color="Maroon" title="c28">a2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q2</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_4"><i><font color="Green" title="E120">A766</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_5_3_4_1_1_2"><i><font color="Green" title="E128">A775</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c13">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_4"><i><font color="Green" title="E121">A767</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_4"><i><font color="Green" title="E122">A768</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23_5_3_4"><i><font color="Green" title="E123">A769</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5_3_4"><i><font color="Green" title="E124">A770</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_4_1_1_2"><i><font color="Green" title="E127">A774</font></i></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></i>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c13">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
<b>by </b><i><a class="txt" href="jordan9.html#E38:33_1_1"><i><font color="Green" title="E61">A421</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_23_5"><i><font color="Green" title="E101">A663</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5"><i><font color="Green" title="E103">A665</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_23_5"><i><font color="Green" title="E106">A668</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E38:33_1_1_23_5"><i><font color="Green" title="E116">A678</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1_23_5"><i><font color="Green" title="E117">A679</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_4"><i><font color="Green" title="E120">A766</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_4_1_1_2"><i><font color="Green" title="E129">A776</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_4_1_1_2"><i><font color="Green" title="E130">A777</font></i></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></i>
;<br/><b>hence </b><a NAME="E6:33_1_1_23_5_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="c16">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_23_5"><i><font color="Green" title="E101">A663</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5"><i><font color="Green" title="E103">A665</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_23_5"><i><font color="Green" title="E107">A669</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_4_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_4_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">p'1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_4_1_1_3"/><b>then </b><i><font color="Green" title="E127">A778</font></i>: 
( <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#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23_5_3_4"><i><font color="Green" title="E123">A769</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5_3_4"><i><font color="Green" title="E124">A770</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_23_5_3_4_1_1_3"/><b>then </b><i><font color="Green" title="E128">A779</font></i>: 
<font color="Maroon" title="c28">a2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q2</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_4"><i><font color="Green" title="E120">A766</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c13">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
<b>by </b><i><a class="txt" href="jordan9.html#E38:33_1_1"><i><font color="Green" title="E61">A421</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_23_5"><i><font color="Green" title="E101">A663</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5"><i><font color="Green" title="E103">A665</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_23_5"><i><font color="Green" title="E106">A668</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E38:33_1_1_23_5"><i><font color="Green" title="E116">A678</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1_23_5"><i><font color="Green" title="E117">A679</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_4"><i><font color="Green" title="E120">A766</font></i></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></i>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c13">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_4"><i><font color="Green" title="E120">A766</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_4"><i><font color="Green" title="E121">A767</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_4"><i><font color="Green" title="E122">A768</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_5_3_4_1_1_3"><i><font color="Green" title="E127">A778</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_4_1_1_3"><i><font color="Green" title="E128">A779</font></i></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></i>
;<br/><b>hence </b><a NAME="E5:33_1_1_23_5_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="c16">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a>, <a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E25:33_1_1_23_5"><i><font color="Green" title="E108">A670</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_4_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_4_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">p'1</font> &amp; <font color="Maroon" title="c28">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p'2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_4_1_1_4"/><b>then </b><i><font color="Green" title="E127">A780</font></i>: 
( <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#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23_5_3_4"><i><font color="Green" title="E123">A769</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5_3_4"><i><font color="Green" title="E124">A770</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_23_5_3_4_1_1_4"/><i><font color="Green" title="E128">A781</font></i>: 
<font color="Maroon" title="c24">p2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q2</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_4"><i><font color="Green" title="E120">A766</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/><a NAME="E4:33_1_1_23_5_3_4_1_1_4"/><i><font color="Green" title="E129">A782</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c11">m</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a></i>;<br/><a NAME="E5:33_1_1_23_5_3_4_1_1_4"/><i><font color="Green" title="E130">A783</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c11">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:33_1_1"><i><font color="Green" title="E42">A95</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a></i>;<br/><a NAME="E6:33_1_1_23_5_3_4_1_1_4"/><i><font color="Green" title="E131">A784</font></i>: 
<font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E7:33_1_1_23_5_3_4_1_1_4"/><i><font color="Green" title="E132">A785</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="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> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E24:33_1_1"><i><font color="Green" title="E51">A351</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E47:33_1_1"><i><font color="Green" title="E67">A427</font></i></a></i>;<br/><a NAME="E8:33_1_1_23_5_3_4_1_1_4"/><i><font color="Green" title="E133">A786</font></i>: 
( <font color="Maroon" title="c28">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c24">p2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">p2</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_23_5_3_4_1_1_4"><i><font color="Green" title="E127">A780</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><i><font color="Green" title="E134">A787</font></i>: <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E50:33_1_1"><i><font color="Green" title="E68">A428</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a></i>
<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">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5_3_4_1_1_4"><i><font color="Green" title="E131">A784</font></i></a></i>
;<br/><a NAME="E10:33_1_1_23_5_3_4_1_1_4"/><i><font color="Green" title="E135">A788</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c11">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">m</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_23"><i><font color="Green" title="E86">A601</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a></i>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c13">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
<b>by </b><i><a class="txt" href="jordan9.html#E38:33_1_1"><i><font color="Green" title="E61">A421</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_23_5"><i><font color="Green" title="E101">A663</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_5"><i><font color="Green" title="E103">A665</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_23_5"><i><font color="Green" title="E106">A668</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E38:33_1_1_23_5"><i><font color="Green" title="E116">A678</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1_23_5"><i><font color="Green" title="E117">A679</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_4"><i><font color="Green" title="E120">A766</font></i></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></i>
<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="c11">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_23_5_3_4"><i><font color="Green" title="E120">A766</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_4"><i><font color="Green" title="E121">A767</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_4"><i><font color="Green" title="E122">A768</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_5_3_4_1_1_4"><i><font color="Green" title="E127">A780</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_4_1_1_4"><i><font color="Green" title="E128">A781</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_4_1_1_4"><i><font color="Green" title="E129">A782</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23_5_3_4_1_1_4"><i><font color="Green" title="E130">A783</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5_3_4_1_1_4"><i><font color="Green" title="E134">A787</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_23_5_3_4_1_1_4"><i><font color="Green" title="E135">A788</font></i></a>, <a class="ref" href="gobrd13.html#T41" title="GOBRD13:th.41">GOBRD13:41</a></i>
;<br/><a NAME="E12:33_1_1_23_5_3_4_1_1_4"/><b>then </b>
<font 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="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E25:33_1_1"><i><font color="Green" title="E52">A352</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23"><i><font color="Green" title="E81">A589</font></i></a>, <a class="txt" href="jordan9.html#E25:33_1_1_23_5"><i><font color="Green" title="E108">A670</font></i></a></i>;<br/><a NAME="E13:33_1_1_23_5_3_4_1_1_4"/><b>then </b>
<font color="Maroon" title="c13">f</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E47:33_1_1"><i><font color="Green" title="E67">A427</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E28:33_1_1_23_5"><i><font color="Green" title="E110">A672</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_5_3_4_1_1_4"><i><font color="Green" title="E132">A785</font></i></a>, <a class="ref" href="gobrd13.html#T45" title="GOBRD13:th.45">GOBRD13:45</a></i>;<br/><a NAME="E14:33_1_1_23_5_3_4_1_1_4"/><b>then </b><i><font color="Green" title="E136">A789</font></i>: 
( <font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c24">p2</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="c23">p1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_23_5"><i><font color="Green" title="E95">A657</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E28:33_1_1_23_5"><i><font color="Green" title="E110">A672</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_5_3_4"><i><font color="Green" title="E121">A767</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_4"><i><font color="Green" title="E122">A768</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5_3_4_1_1_4"><i><font color="Green" title="E133">A786</font></i></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></i>;<br/><a NAME="E15:33_1_1_23_5_3_4_1_1_4"/><b>then </b><i><font color="Green" title="E137">A790</font></i>: 
<font color="Maroon" title="c23">p1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><b>set </b><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="c16">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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span></span>)</span>;<br/><a NAME="E16:33_1_1_23_5_3_4_1_1_4"/><i><font color="Green" title="E138">A791</font></i>: 
<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="c16">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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">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="c16">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_5"><i><font color="Green" title="E91">A653</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5"><i><font color="Green" title="E93">A655</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5"><i><font color="Green" title="E94">A656</font></i></a>, <a class="ref" href="jordan9.html#T29" target="_self" title="JORDAN9:th.29">Th29</a></i>;<br/><a NAME="E17:33_1_1_23_5_3_4_1_1_4"/><i><font color="Green" title="E139">A792</font></i>: 
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>
 <b>by </b><i><a class="txt" href="jordan9.html#E66:33_1_1"><i><font color="Green" title="E79">A585</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E18:33_1_1_23_5_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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E22:33_1_1_23_5"><i><font color="Green" title="E105">A667</font></i></a>, <a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E140">A793</font></i>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4: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="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> = 
<font color="Maroon" title="c16">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan9.html#E17:33_1_1_23_5_3_4_1_1_4"><i><font color="Green" title="E139">A792</font></i></a>, <a class="ref" href="finseq_5.html#T69" title="FINSEQ_5:th.69">FINSEQ_5:69</a></i>
<br/>.= 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E28:33_1_1_23_5"><i><font color="Green" title="E110">A672</font></i></a>, <a class="txt" href="jordan9.html#E17:33_1_1_23_5_3_4_1_1_4"><i><font color="Green" title="E139">A792</font></i></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>
;<br/><span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<font color="Maroon" title="c16">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
<b>by </b><i><a class="txt" href="jordan9.html#E38:33_1_1"><i><font color="Green" title="E61">A421</font></i></a>, <a class="txt" href="jordan9.html#E55:33_1_1"><i><font color="Green" title="E70">A430</font></i></a>, <a class="txt" href="jordan9.html#E57:33_1_1"><i><font color="Green" title="E71">A431</font></i></a>, <a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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>
<b>by </b><i><a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="c16">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>
;<br/><a NAME="E21:33_1_1_23_5_3_4_1_1_4"/><b>then </b>
 <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="c16">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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_23_5"><i><font color="Green" title="E91">A653</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5"><i><font color="Green" title="E93">A655</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_23_5"><i><font color="Green" title="E94">A656</font></i></a>, <a class="txt" href="jordan9.html#E36:33_1_1_23_5"><i><font color="Green" title="E114">A676</font></i></a>, <a class="txt" href="jordan9.html#E37:33_1_1_23_5"><i><font color="Green" title="E115">A677</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_1_23_5_3_4_1_1_4"><i><font color="Green" title="E136">A789</font></i></a>, <a class="txt" href="jordan9.html#E19:33_1_1_23_5_3_4_1_1_4"><i><font color="Green" title="E140">A793</font></i></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></i>;<br/><a NAME="E22:33_1_1_23_5_3_4_1_1_4"/><b>then </b>
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c16">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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1_23_5"><i><font color="Green" title="E119">A681</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_5_3_4"><i><font color="Green" title="E122">A768</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5_3_4"><i><font color="Green" title="E126">A772</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_23_5_3_4_1_1_4"><i><font color="Green" title="E127">A780</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_23_5_3_4_1_1_4"><i><font color="Green" title="E137">A790</font></i></a>, <a class="ref" href="jordan9.html#T22" target="_self" title="JORDAN9:th.22">Th22</a></i>;<br/><a NAME="E23:33_1_1_23_5_3_4_1_1_4"/><b>then </b><i><font color="Green" title="E141">A794</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c16">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="c16">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E27:33_1_1_23_5"><i><font color="Green" title="E109">A671</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E24:33_1_1_23_5_3_4_1_1_4"/><i><font color="Green" title="E142">A795</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c13">f</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_23"><i><font color="Green" title="E87">A605</font></i></a>, <a class="ref" href="sppol_2.html#T22" title="SPPOL_2:th.22">SPPOL_2:22</a></i>;<br/><a NAME="E25:33_1_1_23_5_3_4_1_1_4"/>
 <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="c16">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#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c16">g</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="goboard9.html#D2" title="GOBOARD9:def.2">GOBOARD9:def 2</a></i>;<br/><b>hence </b><a NAME="E26:33_1_1_23_5_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="c16">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_23"><i><font color="Green" title="E89">A633</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_23_5"><i><font color="Green" title="E96">A658</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_5"><i><font color="Green" title="E97">A659</font></i></a>, <a class="txt" href="jordan9.html#E27:33_1_1_23_5"><i><font color="Green" title="E109">A671</font></i></a>, <a class="txt" href="jordan9.html#E16:33_1_1_23_5_3_4_1_1_4"><i><font color="Green" title="E138">A791</font></i></a>, <a class="txt" href="jordan9.html#E23:33_1_1_23_5_3_4_1_1_4"><i><font color="Green" title="E141">A794</font></i></a>, <a class="txt" href="jordan9.html#E24:33_1_1_23_5_3_4_1_1_4"><i><font color="Green" title="E142">A795</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E15:33_1_1_23"/><i><font color="Green" title="E91">A796</font></i>: 
<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="c16">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_6"><b>proof </b></a><div class="add">

<a NAME="E1:33_1_1_23_6"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c13">f</font>,<font color="Maroon" title="c11">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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a>, <a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E47:33_1_1"><i><font color="Green" title="E67">A427</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:33_1_1_23_6"/><i><font color="Green" title="E92">A797</font></i>: 
( <font color="Maroon" title="c18">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="c13">f</font>,<font color="Maroon" title="c11">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> &amp; <font color="Maroon" title="c18">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">p</font> = <font color="Maroon" title="c18">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> <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_23_6"><i><font color="Green" title="E92">A797</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_6_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c20">a</font> = <font color="Maroon" title="c19">p</font>;<br/><b>thus </b><a NAME="E1:33_1_1_23_6_1"/>
<font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_23_6"><i><font color="Green" title="E92">A797</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c21">u</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#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></i>;<br/><b>consider </b><font color="Maroon" title="c22">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> <b> such that </b><br/><a NAME="E4:33_1_1_23_6_1"/><i><font color="Green" title="E93">A798</font></i>: 
<font color="Maroon" title="c22">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>and </b><br/><a NAME="E5:33_1_1_23_6_1"/><i><font color="Green" title="E94">A799</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c21">u</font>,<font color="Maroon" title="c22">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="jordan9.html#E5:33_1_1_23"><i><font color="Green" title="E83">A591</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23"><i><font color="Green" title="E85">A600</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23_6"><i><font color="Green" title="E92">A797</font></i></a>, <a class="ref" href="goboard6.html#T8" title="GOBOARD6:th.8">GOBOARD6:8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">r</font> = <font color="Maroon" title="c22">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><a NAME="E7:33_1_1_23_6_1"/><i><font color="Green" title="E95">A800</font></i>: 
<font color="Maroon" title="c19">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="c23">r</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_23_6_1"><i><font color="Green" title="E93">A798</font></i></a>, <a class="ref" href="goboard6.html#T4" title="GOBOARD6:th.4">GOBOARD6:4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c24">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="c23">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> <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1_23"><i><font color="Green" title="E84">A599</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_1_23_6_1"><i><font color="Green" title="E93">A798</font></i></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></i>;<br/><a NAME="E9:33_1_1_23_6_1"/><i><font color="Green" title="E96">A801</font></i>: 
<font color="Maroon" title="c24">B</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>by </b><i><a class="ref" href="sprect_3.html#T17" title="SPRECT_3:th.17">SPRECT_3:17</a></i>;<br/><a NAME="E10:33_1_1_23_6_1"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c16">g</font>,1,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Maroon" title="c16">g</font>,1
 <b>by </b><i><a class="txt" href="jordan9.html#E58:33_1_1"><i><font color="Green" title="E72">A432</font></i></a>, <a class="txt" href="jordan9.html#E67:33_1_1"><i><font color="Green" title="E80">A586</font></i></a>, <a class="ref" href="gobrd13.html#T34" title="GOBRD13:th.34">GOBRD13:34</a></i>;<br/><a NAME="E11:33_1_1_23_6_1"/><b>then </b><i><font color="Green" title="E97">A802</font></i>: 
 <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="c16">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="c16">g</font>,1</span>)</span>
 <b>by </b><i><a class="ref" href="tops_1.html#T48" title="TOPS_1:th.48">TOPS_1:48</a></i>;<br/><a NAME="E12:33_1_1_23_6_1"/>
 <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="c16">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="c16">g</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E67:33_1_1"><i><font color="Green" title="E80">A586</font></i></a>, <a class="ref" href="goboard9.html#T28" title="GOBOARD9:th.28">GOBOARD9:28</a></i>;<br/><a NAME="E13:33_1_1_23_6_1"/><b>then </b>
 <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="c16">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="c16">g</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E11:33_1_1_23_6_1"><i><font color="Green" title="E97">A802</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E14:33_1_1_23_6_1"/><b>then </b>
 <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="c13">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">g</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E50:33_1_1"><i><font color="Green" title="E68">A428</font></i></a>, <a class="txt" href="jordan9.html#E67:33_1_1"><i><font color="Green" title="E80">A586</font></i></a>, <a class="ref" href="gobrd13.html#T33" title="GOBRD13:th.33">GOBRD13:33</a></i>;<br/><a NAME="E15:33_1_1_23_6_1"/><b>then </b><i><font color="Green" title="E98">A803</font></i>: 
 <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="c13">f</font>,<font color="Maroon" title="c11">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="c16">g</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><a NAME="E16:33_1_1_23_6_1"/><i><font color="Green" title="E99">A804</font></i>: 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c13">f</font>,<font color="Maroon" title="c11">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="c13">f</font>,<font color="Maroon" title="c11">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>
 <b>by </b><i><a class="txt" href="jordan9.html#E42:33_1_1"><i><font color="Green" title="E64">A424</font></i></a>, <a class="txt" href="jordan9.html#E47:33_1_1"><i><font color="Green" title="E67">A427</font></i></a>, <a class="txt" href="jordan9.html#E63:33_1_1"><i><font color="Green" title="E77">A471</font></i></a></i>;<br/><a NAME="E17:33_1_1_23_6_1"/>
<font color="Maroon" title="c24">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="goboard6.html#T6" title="GOBOARD6:th.6">GOBOARD6:6</a></i>;<br/><a NAME="E18:33_1_1_23_6_1"/><b>then </b><i><font color="Green" title="E100">A805</font></i>: 
 <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="c13">f</font>,<font color="Maroon" title="c11">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="c24">B</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_23_6"><i><font color="Green" title="E92">A797</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_1_23_6_1"><i><font color="Green" title="E95">A800</font></i></a>, <a class="txt" href="jordan9.html#E16:33_1_1_23_6_1"><i><font color="Green" title="E99">A804</font></i></a>, <a class="ref" href="tops_1.html#T39" title="TOPS_1:th.39">TOPS_1:39</a></i>;<br/><a NAME="E19:33_1_1_23_6_1"/><i><font color="Green" title="E101">A806</font></i>: 
<font color="Maroon" title="c19">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">B</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_23_6_1"><i><font color="Green" title="E93">A798</font></i></a>, <a class="ref" href="goboard6.html#T4" title="GOBOARD6:th.4">GOBOARD6:4</a></i>;<br/><a NAME="E20:33_1_1_23_6_1"/>
<font color="Maroon" title="c24">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="c16">g</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_23"><i><font color="Green" title="E82">A590</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23_6_1"><i><font color="Green" title="E94">A799</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_23_6_1"><i><font color="Green" title="E96">A801</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_23_6_1"><i><font color="Green" title="E98">A803</font></i></a>, <a class="txt" href="jordan9.html#E18:33_1_1_23_6_1"><i><font color="Green" title="E100">A805</font></i></a>, <a class="ref" href="goboard9.html#T6" title="GOBOARD9:th.6">GOBOARD9:6</a></i>;<br/><b>hence </b><a NAME="E21:33_1_1_23_6_1"/>
<font color="Maroon" title="c20">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="c16">g</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1_23_6_1"><i><font color="Green" title="E101">A806</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:33_1_1_23_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="c16">g</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:33_1_1_23"/>
<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="c16">g</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E14:33_1_1_23"><i><font color="Green" title="E90">A652</font></i></a>, <a class="ref" href="goboard9.html#T27" title="GOBOARD9:th.27">GOBOARD9:27</a></i>;<br/><a NAME="E17:33_1_1_23"/><b>then </b>
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c16">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c16">g</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_1_23"><i><font color="Green" title="E82">A590</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_1_23"><i><font color="Green" title="E83">A591</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_23"><i><font color="Green" title="E91">A796</font></i></a>, <a class="ref" href="jordan9.html#T3" target="_self" title="JORDAN9:th.3">Th3</a></i>;<br/><b>hence </b><a NAME="E18:33_1_1_23"/>
contradiction
 <b>by </b><i><a class="ref" href="sprect_4.html#T7" title="SPRECT_4:th.7">SPRECT_4:7</a></i>;<br/></div><b>end;</b></div>
<b>then </b><i><font color="Green" title="E81">A807</font></i>: <font color="Maroon" title="c16">g</font> = 
<font color="Maroon" title="c13">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 0
<b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>
<br/>.= 
<font color="Maroon" title="c13">f</font>
<b>by </b><i><a class="ref" href="finseq_5.html#T31" title="FINSEQ_5:th.31">FINSEQ_5:31</a></i>
;<br/>
<div><i><font color="Green" title="E82">A808</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_25"><b>now </b></a><div class="add"><b>consider </b><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> <b> such that </b><br/><a NAME="E2:33_1_1_25"/><i><font color="Green" title="E83">A809</font></i>: 
( 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#K1" title="NAT_1:func.1">+</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> )
 <b>and </b><br/><a NAME="E3:33_1_1_25"/><i><font color="Green" title="E84">A810</font></i>: 
(  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 <b>and </b><br/><a NAME="E4:33_1_1_25"/><i><font color="Green" title="E85">A811</font></i>: 
<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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1"><i><font color="Green" title="E35">A80</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c18">i</font> = <font color="Maroon" title="c17">i</font>;<br/><b>thus </b><a NAME="E5:33_1_1_25"/>
( 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#K1" title="NAT_1:func.1">+</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> )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_25"><i><font color="Green" title="E83">A809</font></i></a></i>;<br/><a NAME="E6:33_1_1_25"/><i><font color="Green" title="E86">A812</font></i>: 
<font color="Maroon" title="c13">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
 <b>by </b><i><a class="txt" href="jordan9.html#E24:33_1_1"><i><font color="Green" title="E51">A351</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E45:33_1_1"><i><font color="Green" title="E66">A426</font></i></a></i>;<br/><a NAME="E7:33_1_1_25"/><i><font color="Green" title="E87">A813</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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
 <b>by </b><i><a class="txt" href="jordan9.html#E45:33_1_1"><i><font color="Green" title="E66">A426</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/><a NAME="E8:33_1_1_25"/><b>then </b>
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="c13">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 2</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></i>
<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>
<b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_25"><i><font color="Green" title="E85">A811</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_25"><i><font color="Green" title="E86">A812</font></i></a>, <a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>
;<br/><br/><a NAME="E10:33_1_1_25"/>
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="c13">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_1_25"><i><font color="Green" title="E87">A813</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><font color="Maroon" title="c13">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 2</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2
<b>by </b><i><a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></i>
<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#K1" title="NAT_1:func.1">+</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>
<b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_25"><i><font color="Green" title="E85">A811</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_1_25"><i><font color="Green" title="E86">A812</font></i></a>, <a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>
;<br/><br/><b>thus </b><a NAME="E12:33_1_1_25"/>
(  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_25"><i><font color="Green" title="E84">A810</font></i></a></i>;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c17">f</font> = <font color="Maroon" title="c13">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> <b>by </b><i><a class="txt" href="jordan9.html#E71:33_1_1"><i><font color="Green" title="E81">A807</font></i></a></i>;<br/>
<a NAME="E74:33_1_1"/>
<font color="Maroon" title="c17">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="33_1_1_26"><b>proof </b></a><div class="add">

<a NAME="E1:33_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
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E2:33_1_1_26"/><b>then </b><i><font color="Green" title="E83">A814</font></i>: 
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>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:33_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#K1" title="NAT_1:func.1">+</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">f</font> holds <br/>(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c17">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="c17">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> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a></i>;<br/>
<a NAME="E4:33_1_1_26"/><b>then </b><i><font color="Green" title="E84">A815</font></i>: 
 <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c17">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
 
<b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E72:33_1_1"><i><font color="Green" title="E82">A808</font></i></a>, <a class="ref" href="jordan9.html#T32" target="_self" title="JORDAN9:th.32">Th32</a></i>;<br/>
<a NAME="E5:33_1_1_26"/>
<font color="Maroon" title="c17">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/>
<a NAME="E6:33_1_1_26"/><b>then </b><i><font color="Green" title="E85">A816</font></i>: 
<font color="Maroon" title="c17">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c17">f</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E45:33_1_1"><i><font color="Green" title="E66">A426</font></i></a>, <a class="ref" href="sppol_2.html#T15" title="SPPOL_2:th.15">SPPOL_2:15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">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> <b> such that </b><br/><a NAME="E8:33_1_1_26"/><i><font color="Green" title="E86">A817</font></i>: 
( 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#K1" title="NAT_1:func.1">+</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> )
 <b>and </b><br/><a NAME="E9:33_1_1_26"/><i><font color="Green" title="E87">A818</font></i>: 
<font color="Maroon" title="c17">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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>
 <b>and </b><br/><a NAME="E10:33_1_1_26"/><i><font color="Green" title="E88">A819</font></i>: 
<font color="Maroon" title="c17">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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#K1" title="NAT_1:func.1">+</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>
 <b>and </b><br/><a NAME="E11:33_1_1_26"/>
(  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E72:33_1_1"><i><font color="Green" title="E82">A808</font></i></a></i>;<br/>
<a NAME="E12:33_1_1_26"/><i><font color="Green" title="E89">A820</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">NW-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c17">f</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">NE-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c17">f</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T82" title="PSCOMP_1:th.82">PSCOMP_1:82</a></i>;<br/>
<a NAME="E13:33_1_1_26"/><i><font color="Green" title="E90">A821</font></i>: 
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">NE-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c17">f</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c17">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c17">f</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c17">f</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E14:33_1_1_26"/>
<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> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E8:33_1_1_26"><i><font color="Green" title="E86">A817</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E15:33_1_1_26"/><b>then </b><i><font color="Green" title="E91">A822</font></i>: 
<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="c18">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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_26"><i><font color="Green" title="E83">A814</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_26"><i><font color="Green" title="E86">A817</font></i></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></i>;<br/>
<a NAME="E16:33_1_1_26"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E17:33_1_1_26"/><b>then </b><i><font color="Green" title="E92">A823</font></i>: 
<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="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1_26"><i><font color="Green" title="E83">A814</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_1_26"><i><font color="Green" title="E86">A817</font></i></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></i>;<br/>
<a NAME="E18:33_1_1_26"/>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">NW-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c17">f</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c17">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">NE-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c17">f</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c17">f</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E19:33_1_1_26"/><b>then </b>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">NW-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c17">f</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c17">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K21" title="PSCOMP_1:func.21">NE-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c17">f</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 
<b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1_26"><i><font color="Green" title="E85">A816</font></i></a>, <a class="ref" href="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></i>;<br/>
<a NAME="E20:33_1_1_26"/><b>then </b>
<font color="Maroon" title="c17">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">NW-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c17">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">NE-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c17">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_26"><i><font color="Green" title="E84">A815</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_1_26"><i><font color="Green" title="E87">A818</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_1_26"><i><font color="Green" title="E88">A819</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_1_26"><i><font color="Green" title="E89">A820</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_1_26"><i><font color="Green" title="E90">A821</font></i></a>, <a class="txt" href="jordan9.html#E15:33_1_1_26"><i><font color="Green" title="E91">A822</font></i></a>, <a class="txt" href="jordan9.html#E17:33_1_1_26"><i><font color="Green" title="E92">A823</font></i></a>, <a class="ref" href="goboard7.html#T9" title="GOBOARD7:th.9">GOBOARD7:9</a></i>;<br/>
<a NAME="E21:33_1_1_26"/><b>then </b>
<font color="Maroon" title="c17">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">NW-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c17">f</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">NE-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c17">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#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c17">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E6:33_1_1_26"><i><font color="Green" title="E85">A816</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E22:33_1_1_26"/>
<font color="Maroon" title="c17">f</font> is <a href="sprect_2.html#V1" title="SPRECT_2:attr.1">clockwise_oriented</a>
 <b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_1_26"><i><font color="Green" title="E84">A815</font></i></a>, <a class="ref" href="sprect_2.html#T34" title="SPRECT_2:th.34">SPRECT_2:34</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c18">f</font> = <font color="Maroon" title="c17">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/>
<b>take </b>
<font color="Maroon" title="c18">f</font>
;<br/>

<b>thus </b><a NAME="E76:33_1_1"/>
<font color="Maroon" title="c18">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>
 <b>by </b><i><a class="txt" href="jordan9.html#E19:33_1_1"><i><font color="Green" title="E46">A293</font></i></a></i>;<br/>

<b>thus </b><a NAME="E77:33_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#K1" title="NAT_1:func.1">+</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="c18">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="c18">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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#K1" title="NAT_1:func.1">+</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#K29" title="PSCOMP_1:func.29">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E72:33_1_1"><i><font color="Green" title="E82">A808</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c19">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> ;<br/>

<b>assume </b><a NAME="E78:33_1_1"/><i><font color="Green" title="E83">A824</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">m</font> &amp; <font color="Maroon" title="c19">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c18">f</font> )
 ;<br/>

<a NAME="E79:33_1_1"/>
<font color="Maroon" title="c19">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E80:33_1_1"/><b>then </b><i><font color="Green" title="E84">A825</font></i>: 
<font color="Maroon" title="c19">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E78:33_1_1"><i><font color="Green" title="E83">A824</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E81:33_1_1"/><b>then </b><i><font color="Green" title="E85">A826</font></i>: 
<font color="Maroon" title="c18">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c19">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E24:33_1_1"><i><font color="Green" title="E51">A351</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a></i>;<br/>
<a NAME="E82:33_1_1"/><i><font color="Green" title="E86">A827</font></i>: 
<font color="Maroon" title="c19">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/>
<a NAME="E83:33_1_1"/><i><font color="Green" title="E87">A828</font></i>: 
<font color="Maroon" title="c19">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="jordan9.html#E78:33_1_1"><i><font color="Green" title="E83">A824</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E84:33_1_1"/><i><font color="Green" title="E88">A829</font></i>: 
(  <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="c19">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c19">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="c18">f</font>,<font color="Maroon" title="c19">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> &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> <span class="p2">(<span class="default"><font color="Maroon" title="c19">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c19">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="c18">f</font>,<font color="Maroon" title="c19">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> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E41:33_1_1"><i><font color="Green" title="E63">A423</font></i></a>, <a class="txt" href="jordan9.html#E78:33_1_1"><i><font color="Green" title="E83">A824</font></i></a>, <a class="txt" href="jordan9.html#E80:33_1_1"><i><font color="Green" title="E84">A825</font></i></a>, <a class="txt" href="jordan9.html#E81:33_1_1"><i><font color="Green" title="E85">A826</font></i></a>, <a class="ref" href="gobrd13.html#T43" title="GOBRD13:th.43">GOBRD13:43</a></i>;<br/>
<a NAME="E85:33_1_1"/><i><font color="Green" title="E89">A830</font></i>: 
<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="c19">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">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="c19">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E24:33_1_1"><i><font color="Green" title="E51">A351</font></i></a>, <a class="txt" href="jordan9.html#E39:33_1_1"><i><font color="Green" title="E62">A422</font></i></a>, <a class="txt" href="jordan9.html#E78:33_1_1"><i><font color="Green" title="E83">A824</font></i></a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:33_1_1_27"/>
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c18">f</font>,<font color="Maroon" title="c19">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="c18">f</font>,<font color="Maroon" title="c19">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> )
 ;<br/><a NAME="E2:33_1_1_27"/><b>then </b>
<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="c19">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c19">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>
 <b>by </b><i><a class="txt" href="jordan9.html#E25:33_1_1"><i><font color="Green" title="E52">A352</font></i></a>, <a class="txt" href="jordan9.html#E82:33_1_1"><i><font color="Green" title="E86">A827</font></i></a>, <a class="txt" href="jordan9.html#E83:33_1_1"><i><font color="Green" title="E87">A828</font></i></a>, <a class="txt" href="jordan9.html#E84:33_1_1"><i><font color="Green" title="E88">A829</font></i></a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_27"/>
<font color="Maroon" title="c18">f</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c19">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>
 <b>by </b><i><a class="txt" href="jordan9.html#E78:33_1_1"><i><font color="Green" title="E83">A824</font></i></a>, <a class="txt" href="jordan9.html#E85:33_1_1"><i><font color="Green" title="E89">A830</font></i></a>, <a class="ref" href="gobrd13.html#T44" title="GOBRD13:th.44">GOBRD13:44</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:33_1_1_28"/>
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c18">f</font>,<font color="Maroon" title="c19">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="c18">f</font>,<font color="Maroon" title="c19">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="E2:33_1_1_28"/><b>then </b>
<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="c19">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Maroon" title="c19">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>
 <b>by </b><i><a class="txt" href="jordan9.html#E25:33_1_1"><i><font color="Green" title="E52">A352</font></i></a>, <a class="txt" href="jordan9.html#E82:33_1_1"><i><font color="Green" title="E86">A827</font></i></a>, <a class="txt" href="jordan9.html#E83:33_1_1"><i><font color="Green" title="E87">A828</font></i></a>, <a class="txt" href="jordan9.html#E84:33_1_1"><i><font color="Green" title="E88">A829</font></i></a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_28"/>
<font color="Maroon" title="c18">f</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Maroon" title="c19">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>
 <b>by </b><i><a class="txt" href="jordan9.html#E78:33_1_1"><i><font color="Green" title="E83">A824</font></i></a>, <a class="txt" href="jordan9.html#E85:33_1_1"><i><font color="Green" title="E89">A830</font></i></a>, <a class="ref" href="gobrd13.html#T46" title="GOBRD13:th.46">GOBRD13:46</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E88:33_1_1"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c18">f</font>,<font color="Maroon" title="c19">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="E89:33_1_1"/><b>then </b>
<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="c19">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c19">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>
 
<b>by </b><i><a class="txt" href="jordan9.html#E25:33_1_1"><i><font color="Green" title="E52">A352</font></i></a>, <a class="txt" href="jordan9.html#E82:33_1_1"><i><font color="Green" title="E86">A827</font></i></a>, <a class="txt" href="jordan9.html#E83:33_1_1"><i><font color="Green" title="E87">A828</font></i></a>, <a class="txt" href="jordan9.html#E84:33_1_1"><i><font color="Green" title="E88">A829</font></i></a></i>;<br/>
<b>hence </b><a NAME="E90:33_1_1"/>
<font color="Maroon" title="c18">f</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c19">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>
 <b>by </b><i><a class="txt" href="jordan9.html#E78:33_1_1"><i><font color="Green" title="E83">A824</font></i></a>, <a class="txt" href="jordan9.html#E85:33_1_1"><i><font color="Green" title="E89">A830</font></i></a>, <a class="ref" href="gobrd13.html#T45" title="GOBRD13:th.45">GOBRD13:45</a></i>;<br/>


</div>
